Fractals in Dimension Theory and Complex Networks

BUDAPEST UNIVERSITYOF TECHNOLOGYAND ECONOMICS

DOCTORAL THESIS

Fractals in dimension theory and complex networks

Author: Supervisor: István KOLOSSVÁRY Dr. Károly SIMON

Doctoral School of Mathematics and Computer Science Faculty of Natural Sciences

2019

iii Acknowledgements

It is a great pleasure to thank the many people who have had a direct impact on my academic life in the past years. Foremost, I thank my supervisor Károly Simon. Beyond his deep knowledge of the ﬁeld, his enthusiasm for mathematics and the unique way he gives it on has truly had a great inﬂuence on how I think about math and the world around us;

Secondly, I thank my co-authors Balázs Bárány, Gergely Kiss, Júlia Komjáthy and Lajos Vágó. I gained valuable experience from each and every joint project;

I thank the referees of my home defense for their thorough work. In particular, the comments of István Fazekas, which greatly improved the clarity of one of the chapters.

Moreover, I thank the support and generous hospitality of all the colleagues at the Department of Stochastics. I feel very fortunate to have landed here after my bachelor years, thanks to Doma Szász. The atmosphere is inspiring for research and at the same time very friendly;

and last but not least all the support I get from all my family and friends, who are in some way part of my life, even though many don’t know much about what I actually do on a daily basis.

I acknowledge the ﬁnancial support of different grants and scholarships without which all my travels, meeting many new colleagues, presenting results at conferences and learning many interesting topics would not have been possible.

The template used for the thesis can be found at https://www.latextemplates. com/template/masters-doctoral-thesis. I am very satisﬁed, it saved me from lots of unnecessary headaches.

Contents

Acknowledgements iii

List of Figures vii

List of Symbols ix

List of Acronyms xi

1 Introduction1 1.1 Fractal geometry...... 1 1.2 Self-afﬁne sets...... 2 1.2.1 Planar carpets...... 4 1.2.2 Fractal curves...... 5 1.3 Apollonian networks...... 6 1.4 Informal explanation of contribution...... 8 1.4.1 Dimension of triangular Gatzouras–Lalley-type planar carpets8 1.4.2 Pointwise regularity of zipper fractal curves...... 13 1.4.3 Distances in Random Apollonian Networks...... 15

2 Triangular Gatzouras–Lalley-type planar carpets with overlaps 19 2.1 Triangular Gatzouras–Lalley-type carpets...... 19 2.1.1 Results of Gatzouras and Lalley...... 22 2.1.2 Separation conditions...... 23 Separation of the cylinder parallelograms...... 23 Separation of the columns...... 25 2.2 Main results...... 26 2.2.1 Hausdorff dimension...... 26 2.2.2 Box dimension...... 27 2.2.3 Diagonally homogeneous carpets...... 29 2.3 Preliminaries...... 31 2.3.1 Symbolic notation...... 31 2.3.2 Atypical parallelograms...... 33 2.3.3 Ledrappier–Young formula...... 34 2.4 Upper bound for dimH Λ ...... 35 2.5 Proof of Theorem 2.2.2...... 38 2.5.1 The proof of Theorem 2.2.2 assuming Claim 2.5.1 and Proposi- tion 2.5.2...... 40 2.5.2 The proof of Claim 2.5.1...... 40 2.5.3 Proof of Proposition 2.5.2...... 41 2.6 Proof of results for box dimension...... 45 2.6.1 Diagonally homogeneous subsystems...... 46 2.6.2 Counting intersections...... 49 2.6.3 Proof of Theorem 2.2.7...... 51 vi

2.6.4 Proof of Theorem 2.2.8...... 52 2.7 Examples...... 53 2.7.1 The self-afﬁne smiley: a non diagonally homogeneous example 53 2.7.2 Example for dimH Λ = dimB Λ ...... 53 2.7.3 Overlapping example...... 54 2.7.4 Example "X X"...... 55 ≡ 2.7.5 Negative entries in the main diagonal...... 55 2.7.6 A family of self-afﬁne continuous curves...... 56 2.8 Three-dimensional applications...... 58

3 Pointwise regularity of parameterized afﬁne zipper fractal curves 63 3.1 Self-afﬁne zippers satisfying dominated splitting...... 63 3.2 Main results...... 66 3.3 Pressure for matrices with dominated splitting of index-1...... 68 3.4 Pointwise Hölder exponent for non-degenerate curves...... 73 3.5 Zippers with Assumption A...... 80 3.6 An example, de Rham’s curve...... 85

4 Distances in random and evolving Apollonian networks 89 4.1 Deﬁnitions and notations...... 89 4.2 Main results...... 91 4.2.1 Related literature...... 92 4.3 Structure of RANs and EANs...... 93 4.3.1 Tree-like structure of RANs and EANs...... 93 4.3.2 Distances in RANs and EANs: the main idea...... 97 4.3.3 Combinatorial analysis of shortcut edges...... 98 4.4 Distances in RANs and EANs...... 100 4.4.1 A continuous time branching process...... 100 4.4.2 Proof of Theorem 4.2.1 and 4.2.5...... 103 4.4.3 Proof of Theorem 4.2.3...... 106

A Basic dimension theoretic deﬁnitions 113

B No Dimension Drop is equivalent to Weak Almost Unique Coding 115

Bibliography 119 vii

List of Figures

1.1 A Bedford–McMullen carpet...... 4 1.2 A de Rham curve...... 5 1.3 Linearly parametrized de Rham curve with parameter ω = 1/10....6 1.4 An Apollonian gasket and network...... 7 1.5 Random Apollonian Networks...... 8 1.6 A GL and TGL carpet...... 9 1.7 The "self-affine smiley"...... 9 1.8 Example of Falconer and Miao together with overlapping version... 10 1.9 TGL carpets with different overlaps...... 10 1.10 Example "X X"...... 11 ≡ 1.11 A family of self-affine continuous fractal curves...... 11 1.12 Three-dimensional application...... 12 1.13 Affine zippers in the plane...... 13 1.14 A RAN after a few steps...... 15 1.15 Impact of shortcut edges on diameter of RAN...... 17

2 2.1 The skewness of Ri1...in := fi1...in ([0, 1] ) ...... 21 2.2 The IFS , where z and (1, z) are identiﬁed...... 24 T 2.3 Intersecting parallelograms Rı and R in the proof of Lemma 2.5.4.... 44 2.4 Intersecting parallelograms Rı and R in the proof of Lemma 2.6.8.... 51 2.5 Orientation reversing maps generally destroy the column structure.. 56

3.1 An afﬁne zipper...... 64 3.2 Local neighbourhood of points in Bn,l,m ...... 74 3.3 Well ordered property...... 82

4.1 Coding RANs, initial stpes...... 94 4.2 Coding RANs, induction step...... 94 4.3 Tree like structure of RANs...... 96 4.4 Shortest path from u to v ...... 97

List of Symbols

α (A) ... α (A) singular values of a d d matrix A 1 ≥ ≥ d × α(x), αr(x) pointwise and regular Hölder exponent (3.1.3) , , , Iterated Function Systems F G H T Γ, Λ, Ω attractor of an Iterated Function System Σ, Σ symbolic spaces H Π, Π , Π , π natural projections from a symbolic space s Hs H δ, (δ-approximate) s-dimensional Hausdorff measure (A.0.1) H H2 µd, σd e expectation and variance of Yd (4.1.3) p, q, λ probability vectors p∗ optimal vector for Hausdorff dimension p optimal vector for box dimension µp Bernoulli measure on a symbolic space νep push forward of µp φs singular value function (A.0.4)

D(p) formula for dimH µp (2.2.1) E(β), Er(β) β-level set of α(x) and αr(x) (3.1.5) Id(x) large deviation rate function of Yd (4.1.4) P(t) matrix pressure function (3.2.1) M(x) cone centered at x (3.1.6) Yd full coupon collector block with d + 1 coupons (4.1.2) Diam, Flood, Hop diameter, flooding time, hopcount in a graph (4.1.1) dim an unspecified dimension dimA affinity dimension (A.0.5) dimB, dimB lower and upper box dimension (A.0.3) dimH Hausdorff dimension (A.0.2) dimP packing dimension projx orthogonal projection to x-axis

List of Acronyms

i.i.d. independent and identically distributed ...... 93 u.a.r. uniformly at random ...... 89 w.h.p. with high probability...... 91

AN Apollonian network ...... 7

AUC Almost Uniqe Coding ...... 25

CLT Central Limit Theorem ...... 91

CTBP continuous time branching process ...... 100

EAN evolving Apollonian network ...... 89

GL Gatzouras–Lalley ...... 20

HESC Hochman’s Exponential Separation Condition...... 25

IFS Iterated Function System...... 2

NDD No Dimension Drop ...... 26

OSC Open Set Condition ...... 3

RAN random Apollonian network ...... 89

ROSC Rectangular Open Set Condition ...... 3

SOSC Strong Open Set Condition ...... 3

TGL triangular Gatzouras–Lalley-type ...... 19

WAUC Weak Almost Uniqe Coding...... 25

Chapter 1

Introduction

1.1 Fractal geometry

The word "fractal" comes from the Latin fractus¯ meaning "broken" or "fractured". Benoit B. Mandelbrot coined this term when he wrote in his book Fractals: Form, Chance and Dimension in 1977 that

Many important spatial patterns of Nature are either irregular or frag- mented to such an extreme degree that ... classical geometry ... is hardly of any help in describing their form. ... I hope to show that it is possible in many cases to remedy this absence of geometric representation by using a family of shapes I propose to call fractals – or fractal sets. []

Roughly speaking, any object detailed on arbitrarily small scales that resembles itself in some way on different magnification scales can be called a fractal. Since the 1970s, 80s, fractal geometry has become an important area of mathematics with many connections to theory and practice alike. Fractals have found application in geometric measure theory, dimension theory, dynamical systems, number theory, analysis, differential equations or probability theory to name a few. Also there is increasing interest in more applied areas of mathematics and natural sciences such as network theory, wavelets, percolation problems all the way to computer graphics, image com- pression, financial markets, fluid turbulence or fractal antenna, etc. The list is bound to expand further in the coming years.

Aim and structure of thesis The main aim of the Thesis is to demonstrate this diverse applicability of fractals in different areas of mathematics. Namely,

1. widen the class of planar self-afﬁne carpets for which we can calculate the different dimensions especially in the presence of overlapping cylinders,

2. perform multifractal analysis for the pointwise Hölder exponent of a family of continuous parameterized fractal curves in Rd including deRham’s curve,

3. show how hierarchical structure can be used to determine the asymptotic growth of the distance between two vertices and the diameter of a random graph model, which can be derived from the Apollonian circle packing problem.

Since these topics are not conﬁned to a narrow area of mathematics, a great deal of effort has been put into making the presentation accessible to a wider mathematical audience. 2 Chapter 1. Introduction

This Chapter contains a brief (far from exhaustive) introduction to the topics • and is concluded with a section containing informal explanations of the contri- butions made to the different topics, together with the various methods used in the proofs.

For those not familiar with the terminologies used, AppendixA contains a brief • background material with the basic deﬁnitions and results.

Besides the theoretical results, several examples with illustrative pictures are • provided for explanation.

Plenty of ﬁgures assist the Reader through proofs. • Chapters2,3 and4 contain the precise deﬁnitions and rigorous formulations of our results, together with the proofs. They are based on the papers [KS18; BKK18; KKV16], respectively.

1.2 Self-afﬁne sets

Let (X, d) be a compact metric space. Usually, we will work on a compact subset of Euclidean space Rd. A very natural technique to construct fractals is via Iterated Function Systems (IFSs). An IFS consists of a finite collection of contracting maps F f : X X for i = 1, . . . , N. Huthinson proved in his seminal paper [Hut81], that for i → every IFS there exists a unique non-empty compact set Λ, called the attractor, which satisfies Λ = fi(Λ), i [N] ∈[ where [N] = 1, 2, . . . , N . An extensively studied class of IFSs are the self-affine sets, { } in which case each map of the IFS is an affine transformation, i.e.

fi(x) = Aix + ti, where A Rd d is a contracting, invertible matrix and t Rd is a translation vector. i ∈ × i ∈ An important further subclass consists of the self-similar sets, when the matrices can be written in the form Ai = riOi, where Oi is an orthogonal matrix and 0 < ri < 1 is the contracting ratio in every direction. A natural way to depict an IFS is to provide the images fi(R), where R is the smallest rectangle which contains Λ. Without loss of generality we may assume throughout this thesis that R = [0, 1]2. The correspondence between the IFS and a figure showing the collection of images of R will be unique in our study, since the maps do not contain any rotations or reflections (except in Subsection 2.7.5). See Figure 1.1 for example. Perhaps the most fundamental question in fractal geometry is to determine the dimension of a set. Roughly speaking, dimension indicates how much space a set occupies near to each of its points. Several different types of fractal dimension are used. For basic dimension theoretic definitions such as the Hausdorff, packing and (lower and upper) box dimension of a set and the Hausdorff and local dimension of measures, see AppendixA. Throughout, the Hausdorff, packing, lower and upper box dimension will be denoted by dimH, dimP, dimB, dimB and dimB, respectively. The relative position of cylinder sets fi(R) greatly influence the degree of difficulty to calculate the dimension. The simplest case is the Strong Separation Property (SSP) when all sets fi(Λ) are pairwise disjoint. Somewhat weaker is the (Strong) Open 1.2. Self-affine sets 3

Set Condition. The most difficult is when there is heavy overlapping between the cylinders. This will be central in Chapter2. Definition 1.2.1. An IFS with attractor Λ satisfies the Strong Open Set Condition (SOSC), F if there exists a non-empty open set U, with Λ U = ∅ and such that ∩ 6 f (U) U with f (U) f (U) = ∅ for i = j. (1.2.1) i ⊆ i ∩ j 6 i [N] ∈[ In particular, if U can be chosen to be R, then we say that the Rectangular Open Set Condition (ROSC) holds. This will usually be the case. If U above and Λ can be disjoint, then the Open Set Condition (OSC) holds. In case of self-similar sets satisfying the OSC, all mentioned dimensions are equal to s, often called the similarity dimension, which is the solution of the Hutchinson– Moran formula s ∑ ri = 1, i [N] ∈ see [Fal90, Section 9.2]. Regardless of overlaps, in the self-similar case, the similarity dimension is always an upper bound for the dimensions considered in this thesis. The analog upper bound for self-affine sets is the affinity dimension dimAff, introduced by Falconer [Fal88a], which comes from the "most natural" cover of the set, see (A.0.5) for the definition. All self-affine sets satisfy

dim Λ dim Λ dim Λ min dim Λ, d . H ≤ P ≤ B ≤ { Aff } A central question for the past 30 years has been to determine what can cause the drop of dimension (from the natural upper bounds). An obvious cause is the presence of an exact overlap, i.e. there are two distinct sequences i ,..., i = j ,... j such that 1 n 6 1 k f ... f (Λ) = f ... f (Λ). Another cause in higher dimensions can be the i1 ◦ ◦ in j1 ◦ ◦ jk highly regular alignment of cylinder sets, planar carpets are great examples. The full picture is not completely understood even in the simplest self-similar case on the real line. The transversality method of Pollicott and Simon [PS95], further developed in [Sol95; PS96; PS00] etc., has proven to be a useful tool in determining the dimension of a parametrized family if IFSs for almost all parameter values. Recently, Hochman [Hoc14; Hoc15] made a big breakthrough for the Hausdorff dimension of self-similar measures, which in particular implies that if an IFS on the real line is defined by algebraic parameters, then the drop of dimension is equivalent to having exact overlaps. In a generic sense, equality of dimensions is typical for self-affine sets. Falconer proved in his seminal paper [Fal88a] that for fixed linear parts A ,..., A if A < { 1 N} k ik 1/3 and the translations are chosen randomly according to N d dimensional Lebesgue × measure then all the aforementioned dimensions of the self-affine set are equal. The 1/3 bound was later relaxed by Solomyak [Sol98a] to 1/2, which is sharp due to an example of Przytycki and Urbański[PU89]. Building on the mentioned result of Hochman and results about the Ledrappier–Young formula for self-affine measures [BK17] (see Subsection 2.3.3), very recently Bárány, Hochman and Rapaport [BHR17] greatly improved these results in two dimensions by giving specific, but mild conditions on A ,..., A under which the dimensions are equal. { 1 N} However, in specific cases, which do not fall under these conditions, strict inequality is possible. Planar carpets form a large class of examples in R2 for which this exceptional behavior is typical. The highly regular column and/or row structure causes the drop of the Hausdorff dimension. 4 Chapter 1. Introduction

1.2.1 Planar carpets Independently of each other, Bedford [Bed84] and McMullen [McM84] were the ﬁrst to study planar carpets. They split the unit square R into m columns of equal width and n rows of equal height for some integers n > m 2 and considered iterated ≥ function systems of the form

1/m 0 x i/m f (x) := + (i,j) 0 1/n y j/n for (i, j) A 0, . . . , m 1 0, . . . , n 1 , see Figure 1.1. They gave explicit ∈ ⊆ { − } × { − } formula for the Hausdorff and box-counting dimension of the corresponding attractor Λ. It turns out that dimH Λ = dimB Λ is atypical, namely equality holds if and only if all non-empty columns have the same number of elements.

FIGURE 1.1: A Bedford–McMullen carpet: IFS on left, attractor on right. In this case dimH Λ < dimB Λ.

Later Gatzouras and Lalley [GL92] generalized the results to IFSs, which kept the (non-overlapping) column structure, however the width of the columns could vary and no row structure is required. They still assumed ROSC (1.2.1) and that the width of a rectangle is larger than its height, we say that direction-x dominates. The precise definition is given in Definition 2.1.3, see also Figure 1.6 for an example. More recently, Barańsky[Bar07] kept the row and column structure, but relaxed the direction-x dominates assumption by allowing an arbitrary subdivision of the horizontal and vertical axis. After appropriately choosing which direction is "dominant", the results resemble that of [GL92]. Diagonal systems assuming only ROSC (1.2.1) and no further restrictions on the translations were studied by Feng–Wang [FW05] q and Fraser [Fra12]. Former determined the L spectrum of self-affine measures νp and in particular the box dimension of the attractor. In [Fra12] linear isometries which map [ 1, 1]2 to itself are allowed and the box dimension is determined. Fraser called − these box-like sets. Observe that in all the mentioned papers the ROSC (1.2.1) was assumed. Carpets with overlaps were not studied until the last few years. Fraser and Shmerkin [FS16] shift the columns of Bedford–McMullen carpets to get overlaps, while Pardo-Simón [PS] allows shifts in both directions of Barańskicarpets. Relying on a recent breakthrough by Hochman [Hoc14] on the dimension of self-similar measures on the line, both papers show that apart from a small exceptional set of parameters the results in [Bed84; McM84] and [Bar07] remain valid in the overlapping case. This is the type of shifted columns that can be seen in Figure 1.9. 1.2. Self-affine sets 5

In Chapter2 triangular Gatzouras–Lalley carpets are introduced: the column structure of Gatzouras–Lalley carpets are kept, however instead of diagonal matrices, lower triangular ones are used to define the linear parts of the maps. Hence, the usual rectangular cylinder sets become parallelograms with two vertical sides, see Figure 1.6. Moreover, we also allow different types of overlaps in the construction, see Figure 1.9. The results of [GL92] are generalized to this setting. The parallelograms and overlaps give a much more flexible framework in which existing and many new examples can be treated in a unified way.

1.2.2 Fractal curves Fractal curves have already appeared in the 19th century, although at the time the un- usual constructions were considered by many to be mathematical "monsters". Peano created the first space-filling curve, Weierstrass presented the first example for a continuous but nowhere differentiable function and von Koch gave a more geometric definition of a fractal curve simply referred to today as the von Koch snowflake. Since then, fractal curves have found abundant applications in areas such as wavelets, interpolation functions or signal processing. The starting point of our work was the curve introduced and studied by de Rham [Rha47; Rha56; Rha59]. The geometric construction of the curve goes as follows. Starting from the square [0, 1] [ 1, 0], it can be obtained by trisecting each side with ratios ω : (1 2ω) : ω × − − and "cutting the corners" by connecting each adjacent partitioning point to get an octagon. Again, each side is divided into three parts with the same ratio and adjacent partitioning points are connected, and so on. The de Rham curve is the limit curve of this procedure. With a more analytic approach, in the language of iterated function systems we can say that de Rham’s curve is the attractor Γ of the IFS

ω 0 0 1 2ω ω 2ω f0(x) = x and f1(x) = − x + , (1.2.2) ω 1 2ω − 2ω 0 ω 0 − where ω (0, 1/2) is the parameter. More precisely, the self-affine curve Γ defined ∈ by (1.2.2) gives the segment between two midpoints of the original square. Observe that the fixed point of f is z = [0, 1]T, while for f it is z = [1, 0]T. Furthermore, 0 0 − 1 2 f0(z2) = z1 = f1(z0), we say that the cross-condition holds. This ensures that Γ is continuous. Figure 1.2 shows both approaches: first four steps of the "corner cutting" on left and the images of f0 and f1 after one (red), two (green) and three (black) levels of iteration.

FIGURE 1.2: A de Rham curve: geometric construction (left), IFS construction (right). 6 Chapter 1. Introduction

The main goal of Chapter3 is to analyze the pointwise regularity of such self- affine curves given by a linear parametrization in the much more general framework of zipper IFSs as defined in [ATK03], see Definition 3.1.1. In case of de Rham’s curve, a linear parametrization v : [0, 1] R2 is of the form 7→ i i + 1 v(x) = f (v(2x i)), for x , , i = 0, 1. (1.2.3) i − ∈ 2 2 h Figure 1.3 shows a visualisation of a linearly parametrized de Rham curve. Such linear parameterizations occur in the study of wavelet functions in a natural way, see for example Protasov [Pro06], Protasov and Guglielmi [PG15], and Seuret [Seu16].

FIGURE 1.3: Linearly parametrized de Rham curve with parameter ω = 1/10. Left: The image of the unit cube w.r.t the IFS generating the graph of the de Rham curve. Middle: The second iteration. Right: The curve itself.

The analysis is done by performing multifractal analysis, i.e. to study the possible values, which occur as (regular) pointwise Hölder exponents, and determine the mag- nitude of the sets (in terms of Hausdorff dimension), where it appears. This property was studied for several types of singular functions, for example for wavelets by Barral and Seuret [BS05], Seuret [Seu16], for Weierstrass-type functions by Otani [Ota17], for complex analogues of the Takagi function by Jaerisch and Sumi [JS17] or for different functional equations by Coiffard, Melot and Willer [CMT14], by Okamura [Oka16] and by Slimane [BS03] etc. Our results applied to de Rham’s curve give ﬁner results than existing ones in the literature.

1.3 Apollonian networks

Since the seminal work of Erd˝osand Rényi [ER60], the theory of random graphs has received immense interest in many areas of science. Social networks, the World Wide Web, traffic/shipping routes etc. have made it essential to create models which cap- ture the main features and driving forces of these real-world networks, often also called complex networks. These main features include large clustering coefficient, power law decay of degree distribution and small distances between nodes (logarithmic with size) meaning that networks tend to contain smaller, dense communities; there are few really large "hubs" and many-many much smaller ones; finally "it’s a small world after all". A particularly successful class of models are the Preferential attachment models [BA99; BR04; Bol+01; FP16]. In these dynamically evolving graphs, the probability that a new node attaches to an existing one is proportional to the degree of the existing one. 1.3. Apollonian networks 7

Thus, nodes with already large degree tend to get new edges more frequently, hence the term preferential attachment. This simple mechanism gives rise to the power law decay. Several variants of the model have been deﬁned, here we focus on the shortest paths between vertices of Apollonian networks. The construction of deterministic and random Apollonian networks originates from the problem of Apollonian circle packing: starting with three mutually tangent circles, we inscribe in the interstice formed by the three initial circles the unique circle that is tangent to all of them: this fourth circle is known as the inner Soddy-circle. It- eratively, for each new interstice its inner Soddy-circle is drawn, see Figure 1.4. After inﬁnite steps the result is a fractal set, an Apollonian gasket [Boy82; Gra+03]. An Apollonian network (AN) is the resulting graph if we place a vertex in the center of each circle and connect two vertices if and only if the corresponding circles are tangent, see Figure 1.4. This model was introduced independently by Andrade et al. [And+05] and Doye and Massen [DM05] as a model for real-world networks. These networks have the important properties mentioned above, in addition, due to the construction, Apollonian networks also show hierarchical structure: another property very commonly observed.

FIGURE 1.4: An Apollonian gasket (left), the deterministic network generated by it (right).

It is straightforward to generalize Apollonian packings to Rd for d 2 with ≥ mutually tangent d dimensional hyperspheres. Analogously, if each d-hypersphere corresponds to a vertex and vertices are connected by an edge if the corresponding d-hyperspheres are tangent, then we obtain a d-dimensional AN (see [Zha+08; Zha+06]). The network arising by this construction is deterministic. Zhou et al. [ZYW05] proposed to randomize the dynamics of the model such that in one step only one interstice is picked uniformly at random and filled with a new circle. This construction yields a d dimensional random Apollonian network (RAN) [ZRC06], see Figure 1.5. Using heuristic and rigorous arguments the results in [AM08; CFU14; DS07; Ebr+13; FT12; ZRC06; ZYW05] show that RANs have the above mentioned main features of real-world networks. A different random version of the original Apollonian network was introduced by Zhang et al. [ZRZ06], called Evolutionary Apollonian networks (EAN) where in every step every interstice is picked and filled independently of each other with probability q. If an interstice is not filled in a given step, it can be filled in the next step again. We call q the occupation parameter. For q = 1 we get back the deterministic AN model. It is 8 Chapter 1. Introduction

FIGURE 1.5: An array of Random Apollonian Networks in two dimensions with n = 50, 250, 750 vertices. conjectured in [ZRZ06] that an EAN with parameter q, as q 0, should show similar → topological behaviour to RANs. To make this statement rigorous, instead of looking at a sequence of evolving EAN-s with decreasing parameters, we slightly modify the model and investigate the asymptotic behaviour of a single EAN when q might differ in each step of the dynamics. That is, we consider a series q ∞ of occupation { n}n=1 parameters so that qn applies for step n of the dynamics, and assume that qn tends to 0. In this setting, the interesting question is to determine the correct rate for qn that achieves the observation that EAN shows similar behaviour as RAN when the parameter tends to zero. The main goal of Chapter4 is to determine the precise asymptotic growth of the shortest path between two vertices and the diameter of the graph.

1.4 Informal explanation of contribution

The main body of the Thesis is based on three articles [KS18; BKK18; KKV16]. We now informally explain the contribution and methods used in each of these papers.

1.4.1 Dimension of triangular Gatzouras–Lalley-type planar carpets Gatzouras–Lalley (GL) carpets [GL92] are the attractors of self-affine IFSs on the plane whose first level cylinders are aligned into columns using orientation preserving maps with linear parts given by diagonal matrices. In Chapter2, we consider a natural generalization of such carpets by replacing the diagonal matrices with lower triangular ones so that the column structure is preserved, see Definition 2.1.1. We call them Triangular Gatzouras–Lalley-type (TGL) planar carpets, indicating that the linear part of the maps defining the IFS are triangular matrices and it is a natural generalization of the Gatzouras–Lalley construction. The shaded rectangles and parallelograms in Figure 1.6 show the images of R under the maps defining a GL carpet on the left and a TGL carpet on the right. These are typical examples which satisfy the ROSC (1.2.1). Furthermore, there is a correspondence between the rectangles and parallelograms so that the height and width of corresponding ones coincide. We call the Gatzouras–Lalley carpet the GL-brother of the TGL carpet, see Definition 2.1.4. Even though the ROSC holds, it is not immediate that the dimension of the two attractors should be the same. The parallelograms can be placed in a way that there is no bi-Lipschitz map between the two attractors. Nevertheless, Barańskiessentially shows in [Bar08] that assuming the ROSC the 1.4. Informal explanation of contribution 9

Hausdorff and box dimension of a TGL carpet is equal to the respective dimension of its GL brother.

FIGURE 1.6: The IFS deﬁning a Gatzouras–Lalley carpet on left and triangular Gatzouras–Lalley-type carpet on right, which are brothers.

The IFS in Figure 1.6 is an example for which dimH Λ < dimB Λ < dimAff Λ. If the orthogonal projection of Λ to the x-axis is the whole [0, 1] interval, then the box- and affinity dimensions are equal. Figure 1.7 shows such an example, where the outlines of fi(R) are shown together with the attractor, which we call the "self-affine smiley". A special class of examples consists of affine IFSs in which all matrices have the same main diagonal. We call them diagonally homogeneous carpets. Such a particular TGL carpet (see the left hand side of Figure 1.8) was introduced by Falconer and Miao [FM07, Figure 1 (a)], where the box dimension of the attractor was calculated. Later, Bárány [B15´ , Subsection 4.3] showed that for this example the box and the Hausdorff dimensions are the same. This is due to the fact that all columns have the same number of maps. The well-known Bedford–McMullen carpets [Bed84; McM84] form a proper subclass of these TGL carpets. In all these examples only the boundary of the cylinder sets fi(R) could intersect. However, the main contribution of Chapter2 is to continue the works [Bar08; GL92] to allow different types of overlaps in the construction. Figure 1.9 illustrates the overlaps we consider. All three examples are brothers of the GL carpet in Fig- ure 1.6. On the left, the columns are shifted in a way that the IFS on the x-axis generated by the columns, denoted by , satisfies Hochman’s Exponential Separation H Condition (HESC), see Definition 2.1.9. This type of shifted columns were considered

FIGURE 1.7: The "self-afﬁne smiley", whose dimH = 1.20665 . . . < 1.21340 . . . = dimB = dimA, see Subsection 2.7.1. 10 Chapter 1. Introduction

FIGURE 1.8: The attractor Λa from Subsection 2.7.2 with parameter a = 3/10 (left) and Subsection 2.7.3 with parameter a = 3/20 (right), shown together with the outlines of the images of fi(R).

by Fraser and Shmerkin [FS16] and Pardo-Simón [PS] on different carpets. In the center, columns do not overlap, however, parallelograms within a column may do so if a certain transversality like condition holds. The one on the right on Figure 1.9 has both types of overlaps.

FIGURE 1.9: Triangular Gatzouras–Lalley-type carpets with different overlaps. Left: shifted columns satisfying Hochman’s exponential separation condition. Center: non-overlapping columns, transversality condition. Right: mixture of both.

By modifying the translation vectors in the example on the left hand side of Fig- ure 1.8, we got a brother with overlaps seen on the right hand side, for which we show in Subsection 2.7.3 that transversality holds. Another concrete overlapping example satisfying transversality is "X X" in Figure 1.10, for which there is strict ≡ inequality between the Hausdorff, box and afﬁnity dimensions. If instead the construction would be "X = X", then the Hausdorff and box dimensions would be equal. Moreover, if there were no empty columns in this example, then the box and afﬁnity dimensions would coincide.

Results Section 2.2 contains the formal statements of all the main results. Roughly speaking, we show that for any TGL carpet Λ

dim Λ dim Λ, ≤ e 1.4. Informal explanation of contribution 11

FIGURE 1.10: Example "X X" from Subsection 2.7.4 for which ≡ dimH Λ = 1.13259 . . . < dimB Λ = 1.13626 . . . < dimA Λ = 1.2170 . . . .

where Λ is the GL brother of Λ and dim means either box or Hausdorff dimension, see Theorems 2.2.1 and 2.2.4. When ROSC holds and the IFS generated by the H columnse satisfies Hochman’s condition, then equality can be deduced from recent works [BK17; Fra12]. Also, equivalent conditions are given for the equality of the different dimensions. Our main contribution is that in the presence of overlaps described above, we give sufficient conditions under which dim Λ does not drop below dim Λ, see Theo- rems 2.2.2 and 2.2.7. In particular, for the Hausdorff dimension we allow both types of overlaps simultaneously (like the third figure in Figure 1.9), howevere for the box dimension it is either one or the other type (like the first two in Figure 1.9). For a discussion on generalizing towards orientation reversing maps, see Subsec- tion 2.7.5. In particular, we calculate the dimension of a family of self-affine continuous curves Λ , which is generated by an IFS containing a map that reflects on the a Fa y-axis, see Figure 1.11. They are examples for the type of fractal curves we study from a different perspective in Chapter3. The formal treatment of all the mentioned examples is done in Section 2.7.

FIGURE 1.11: Left: ﬁrst (red), second (green) and third (black) level cylinders of a. Right: Λa rotated 90 degrees from Subsection 2.7.6 with parameterF a = 0.2 (black), 0.12 (red) and 0.08 (green).

One motivation to study self-afﬁne fractals of overlapping construction is that 12 Chapter 1. Introduction

sometimes the dimension of a higher dimensional fractal of non-overlapping construction coincides with its lower dimensional orthogonal projection which can be a self-affine fractal of overlapping construction. We obtain such a set in 3D by starting from a TGL carpet with overlaps on the xy-plane and then "lift" it to 3D so that the in- teriors of the first level cylinders are disjoint. Figure 1.12 shows such an example with the first level cylinders (left), the attractor (center) and the projection of the cylinders and attractor to the xy-plane (right). Section 2.8 contains the formal treatment of this type of construction.

FIGURE 1.12: A three-dimensional fractal whose dimension is equal to the dimension of its orthogonal projection to the xy-plane.

Methods used to handle overlaps For each type of overlap and dimension we used different methods:

The upper bounds on the Hausdorff and box dimensions (after some simple ob- • servations) follow from proper adaptations of the results of Gatzouras–Lalley [GL92] and Fraser [Fra12], respectively.

To estimate the Hausdorff dimension from below we use the Ledrappier-Young • formula of Bárány and Käenmäki [BK17] (cited in Theorem 2.3.4) for self-afﬁne measures. We show that this lower bound equals the upper bound

– in case of overlapping like on the second ﬁgure of Figure 1.9 by an argument inspired by the transversality method introduced in [BRS]; – in case of overlapping like on the third ﬁgure of Figure 1.9, we introduce a new separation condition for the self-similar IFS obtained as the projection of the TGL carpet under consideration to the horizontal line. This separation condition is a non-trivial consequence of Hochman’s Exponential Separation Condition [Hoc14]. We prove this in AppendixB since it could be of separate interest.

To estimate the box dimension from below we could not simply use the Haus- • dorff dimension of the attractor, because, in our case, it is (typically) strictly smaller than the box dimension. Therefore,

– in case of overlapping like on the ﬁrst ﬁgure of Figure 1.9, we used the method of Fraser and Shmerkin [FS16]: the main idea is to pass to a special subsystem of a higher iterate of the IFS which has non-overlapping columns; 1.4. Informal explanation of contribution 13

– in case of overlapping like on the second ﬁgure of Figure 1.9, we introduced a new argument to count overlapping boxes. It uses transversality and a result of Lalley [Lal88] based on renewal theory, which gives the precise asymptotics of the number of boxes needed to cover the projection of the attractor to the horizontal line.

1.4.2 Pointwise regularity of zipper fractal curves We postpone the general definition of zipper fractal curves, as given in [ATK03], until Chapter3. For now it is enough to consider a self-affine IFS = f0,..., fN 1 F { − } in which all matrices have strictly positive entries and map [0, 1]2 into itself. Fur- T T thermore, the fixed points of f0 and fN 1 are [0, 0] and [1, 1] , respectively, and T T − fi([1, 1] ) = fi+1([0, 0] ). Figure 1.13 shows such examples in the plane with the first (red), second (green) and third (black) level cylinders visible.

FIGURE 1.13: Afﬁne zippers in the plane deﬁned by matrices with strictly positive entries.

Let λ = (λ0,..., λN 1) be a probability vector, which defines a subdivision of − [0, 1]. Let gi be the affine function mapping the unit interval [0, 1] to the ith subinterval of the division (indexing from 0 to N 1). Then a linear parametrization of the attractor − Γ of is a function v : [0, 1] Rd defined by the functional equation F → 1 v(x) = f v(g− (x)) if x g ([0, 1]). i i ∈ i In comparison, we call F : Rd R a self-similar function if there exists a bounded 7→ open set U Rd, and contracting similarities g ,..., g of Rd such that g (U) ⊂ 1 k i ∩ g (U) = ∅ and g (U) U for every i = j, and a smooth function g : Rd R, j i ⊂ 6 7→ and real numbers ρ < 1 for i = 1, . . . , k such that | i| k 1 F(x) = ∑ ρi F(gi− (x)) + g(x), (1.4.1) i=1 see [Jaf97b, Definition 2.1]. The graph of F over the attractor of g ,..., g can be { 1 k} written as the unique, non-empty, compact invariant set of the family of functions d+1 S1,..., Sk in R , where

Si(x, y) = (gi(x), ρiy + g(gi(x))). 14 Chapter 1. Introduction

The multifractal formalism of the pointwise Hölder exponent of self-similar functions was studied in several aspects, see for example Aouidi and Slimane [AS04], Slimane [BS01; BS03; BS12] and Saka [Sak05]. The main difference between the self-similar function F defined in (1.4.1) and v defined in (3.1.2) is the contraction part. Namely, while F is a real valued function rescaled by only a real number, the function v is Rd valued and a strict affine transformation is acting on it. This makes the study of such functions more difficult. When studying the pointwise Hölder exponent

log v(x) v(y) α(x) = lim inf k − k, y x log x y → | − | we need a good control over the distance v(x) v(y) as y x. This is much k − k → simpler for the self-similar function F, since F was scaled only by a constant. Roughly speaking

F(g (x)) F(g (y)) ρ ρ F(x) F(y) . k i1,...,in − i1,...,in k ≈ | i1 ··· in |k − k Whereas, in the case of self-afﬁne systems, this is not true anymore. That is,

v(g (x)) v(g (y)) A A (v(x) v(y)) . k i1,...,in − i1,...,in k ≈ k i1 ··· in − k However, in general A A (v(x) v(y)) A A v(x) v(y) . In order k i1 ··· in − k 6≈ k i1 ··· in kk − k to be able to compare the distance v(g (x)) v(g (y)) with the norm of the k i1,...,in − i1,...,in k product of matrices, we need an extra assumption on the family of matrices. This notion is called dominated splitting, see Deﬁnition 3.1.2. The example of matrices with strictly positive entries satisﬁes this condition.

Results The key technical tool is the matrix pressure function P(t) defined in (3.2.1). Under a mild non-degeneracy condition (3.1.7) and dominated splitting we show that α(x) is equal to a constant α = P (0) for Lebesgue almost all x [0, 1]. Furthermore, the 0 ∈ multifractal formalism holds for the Hausdorff dimension of the level sets E(β) = x [0, 1] : α(x) = β of a linear parametrization of an affine zipper in a small { ∈ b } interval of possible β values. In other words dimH E(β) is equal to the Legendre- transform of P(t), see Theorem 3.2.1. Moreover, an important contribution was to find an additional rather natural assumption, see Assumption A (3.2), which is equivalent for the lim inf in the definition of α(x) to exist as a limit for Lebesgue almost every point, see Theorem 3.2.3. We call the limit the regular Hölder exponent and denote it αr(x). Under these conditions Theorem 3.2.2 states that the multifractal formalism for αr(x) can be extended to the full spectrum of possible β values. For example, de Rham’s curve and matrices with strictly positive entries satisfy Assumption A, hence these results can be applied to get stronger results than the ones in the literature. In general, our results do not imply differentiability of the linear parametrization v, because the lower bound we obtain for α can be strictly smaller than 1. However, in the particular case of de Rham’s curve we show in Proposition 3.6.2 that v is differentiable for Lebesgue-almost every x withb derivative vector equal to zero and the set where it is not differentiable has strictly positive Hausdorff dimension. 1.4. Informal explanation of contribution 15

Unfortunately, even in the simplest case of positive matrices, there is not much hope for the computation of the precise values of P(t). Only fast approximation algorithms exist, see works of Pollicott and Vytnova [PV15] and Morris [Mor18].

Methods The most important technical contribution was to generalize the results of Feng [Fen03], and Feng and Lau [FL02] for the pressure function of infinite products of positive matrices to the more general dominated splitting setting. This is based on results by Bochi and Gourmelon [BG09] characterizing systems with dominated splitting and general ergodic theoretic machinery of Bowen [Bow08]. The proof of the main results use the properties of the pressure function and rely on handling the points which are far away symbolically but close on the self-affine curve, which is done by a fine study of the underlying dynamical system associated with the matrices Ai.

1.4.3 Distances in Random Apollonian Networks We postpone the precise deﬁnition of RANs and EANs in arbitrary dimensions until Chapter4, instead just focus on RANs in two dimensions. There is a natural representation of RANs as evolving triangulations in two dimensions: take a planar embedding of the complete graph on four vertices as in Fig- ure 1.14 and in each step pick a face of the graph uniformly at random, insert a vertex and connect it with the vertices of the chosen triangle (face). The result is a maximal planar graph. Hence, a (RAN2(n))n N is equivalent to an increasing family of ∈ triangulations by successive addition of faces in the plane, called stack-triangulations. Stack-triangulations were investigated in [AM08] where the authors also considered typical properties under different weighted measures, (e.g. ones picked u.a.r. having n faces). Under a certain measure stack-triangulations with n faces are an equivalent formulation of RAN2(n), see [AM08] and references therein.

FIGURE 1.14: A RAN2(n) after n = 0, 2, 8 steps

Let u and v be two vertices of a RAN with n vertices. To study distances we denote by Hop(n, u, v) the hopcount between the vertices, i.e., the number of edges on (one of) the shortest paths between u and v. The flooding time Flood(n, u) is the maximal hopcount from u, while the diameter Diam(n) is the maximal flooding time (4.1.1). The terminology comes from the intuitive picture that if we were to open a source of fluid at u that spreads with unit speed along the edges, then the flooding time is exactly the time when the fluid reaches all other vertices of the graph. 16 Chapter 1. Introduction

Results We determine the exact asymptotic growth of Hop, Flood and Diam as the number of vertices tend to infinity. The results are valid in all dimensions d and all constants are explicit, only depending on d. All three quantities scale as c log n for different constants, which confirms the small-world property. More precisely, we show that for both RANs and EANs the hopcount obeys a central limit theorem (CLT), when the two vertices are chosen randomly according to a well-defined probability distribution. The centralizing constant is of order log n while the normalizing one is of order log n, see Theorems 4.2.1 and 4.2.5. Furthermore, for RANs the flooding time and diameter divided by log n tend in probability to well p defined constants, see Theorem 4.2.3. The constants that arise have very intuitive meaning once the structure of these graphs is understood. Understanding the hierarchical structure using the fractal viewpoint is perhaps the most important contribution of this work. It gave the opportunity to analyze all the quantities simultaneously, independently of the dimension in a unified way. The CLT for the hopcount is novel, it is especially interesting, because to the best of our knowledge in all previous cases, when a CLT was proved, the underlying graph model had random edgeweights, while RANs and EANs do not. Determining the diameter is always a greater challenge than the hopcount. The flooding time is very rarely studied, in particular for RANs it was not known before. The exact relation of our results compared to existing ones in the literature is surveyed in Subsection 4.2.1. We remark that the article [KKV16] where these results appeared also contain results about the degree distribution and clustering coefficient, however they are not presented in this thesis.

Methods The proofs rely on a few key observations. The first is to realize that RANs have a nice tree-like structure: each vertex can be assigned a unique finite code and all edges can be grouped into two categories, either tree edges or shortcut edges. The tree edges go "down" in the hierarchy of the graph, while a shortcut edge goes back "upward" toward the initial graph along a branch of the tree. See Figure 4.3 for an illustration of the graph in Figure 1.14. All neighbors of a vertex higher in the hierarchy can be determined simply by looking at its unique code. This allows a combinatorial analysis and the well-known coupon collector problem comes up naturally. The idea of using codes can be attributed to my co-author Lajos Vágó, which was then further developed. Secondly, there is a natural embedding of the evolution of a RAN into the evolution of a continuous time branching process (CTBP). This allows for the application of results for CTBPs [AN04; BD06; Büh71] to determine the average and maximal depth of the tree created by the tree edges. The CLT for the hopcount can be derived from here. However, for the flooding time and diameter an extra important observation is needed. Due to the shortcut edges, the deepest branch of the tree created by the tree edges is not necessarily the farthest from the initial graph in graph distance. There can be many branches slightly shorter, however farther from the initial graph, because the shortcut edges climb to the top slower. Figure 1.15 illustrates this phenomena. To determine the exact constants, we achieve the maximal distance by an entropy vs energy argument. 1.4. Informal explanation of contribution 17

branch farthest in graph distance

branch with most edges

other branches

shortcut edge

FIGURE 1.15: Impact of shortcut edges on diameter of RAN: branch with most edges is not the one farthest in graph distance.

The precise proofs use techniques from renewal theory, CTBP theory, large deviations, second moment method, symbolic dynamics and combinatorics.

Chapter 2

Triangular Gatzouras–Lalley-type planar carpets with overlaps

This chapter is based on [KS18] written jointly with Károly Simon. Recall Figure 1.6.

2.1 Triangular Gatzouras–Lalley-type carpets

Denote the closed unit square by R = [0, 1] [0, 1]. Let = A ,..., A be a family × A { 1 N} of 2 2 invertible, strictly contractive, real-valued lower triangular matrices. The × corresponding self-afﬁne IFS is the collection of afﬁne maps

N bi 0 ti,1 = fi(x) := Aix + ti i=1, where Ai = and ti = , (2.1.1) F { } di ai ti ,2 for translation vectors t , with t , t 0. We assume that a , b (0, 1). i i,1 i,2 ≥ i i ∈ Orthogonal projection of to the horizontal x-axis, denoted proj , generates an F x important self-similar IFS on the line

= h (x) := b x + t N . (2.1.2) H { i i i,1}i=1 We denote the attractor of eand eby Λ = Λ and Λ respectively. F H F H Deﬁnition 2.1.1. We say that an IFSe of the form (2.1.1) isetriangular Gatzouras–Lalley- type (TGL) and we call its attractor Λ a TGL planar carpet if the following conditions hold: (a) direction-x dominates, i.e.

0 < a < b < 1 for all i [N] := 1, 2, . . . , N , (2.1.3) i i ∈ { } (b) column structure: there exists a partition of [N] into M > 1 sets ,..., with cardi- I1 IM nality = N > 0 so that |Iıˆ| ıˆ

1 = 1, . . . , N1 and ıˆ = N1 + ... + Nıˆ 1 + 1, . . . , N1 + ... + Nıˆ (2.1.4) I { } I { − } for ıˆ = 2, . . . , M. Assume that for two distinct indices k and ` 1, . . . , N ∈ { }

bk = b` =: rıˆ, if there exists ıˆ 1, . . . , M such that k, ` ıˆ, then (2.1.5) ∈ { } ∈ I (tk,1 = t`,1 =: uıˆ.

We also introduce = h (x) := r x + u M , (2.1.6) H { ıˆ ıˆ ıˆ}ıˆ=1 and we observe that the attractor Λ of is identical with Λ . H H H e 20 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

u + r u for ıˆ = 1, . . . , M 1 and u + r 1. (2.1.7) ıˆ ıˆ ≤ ıˆ+1 − M M ≤ (d) Without loss of generality we always assume in this paper that

(A1) f (R) R for all i [N] and i ⊂ ∈ (A2) The smallest and the largest ﬁxed points of the functions of are 0 and 1 respec- H tively.

Observe that the definition allows overlaps within columns (like the second figure in Fig- ure 1.9), but columns do not overlap. We say that Λ is a shifted TGL carpet if we drop the assumption (2.1.7), that is non- M overlapping column structure is NOT assumed, we require only that ∑ rıˆ 1 (like the ıˆ=1 ≤ first figure in Figure 1.9).

We often consider the following special cases:

Deﬁnition 2.1.2. We say that a shifted TGL carpet Λ has uniform vertical fibres if

s s ∑ aj− H = 1 for every ıˆ [M], (2.1.8) j ∈ ∈Iıˆ where s = dimB Λ and s = dimB Λ . H H Furthermore, we call Λ a diagonally homogeneous shifted TGL carpet if

b b and a a for every i [N]. i ≡ i ≡ ∈ In particular, a diagonally homogeneous carpet has uniform vertical ﬁbres if N/M N and ∈ N = N/M for every ıˆ 1, . . . , M . ıˆ ∈ { } The special case when Nıˆ = 1 for all ıˆ = 1, . . . , M is treated in the paper of Bárány, Rams and Simon [BRS, Lemma 3.1]. GL carpets are just special cases of TGL carpets.

Definition 2.1.3. A self-affine IFS is a Gatzouras–Lalley (GL) IFS and its attractor Λ is F a GL carpet if is a TGL IFS as in Definition 2.1.1 with the additional assumptions that F all off-diagonal elements di = 0 ande the rectangular open set condition (ROSC) holds, recalle Definition 1.2.1.e

Deﬁnition 2.1.4. Let Λ be a shifted TGL carpet generated from the IFS of the form (2.1.1). F We say that the Gatzouras–Lalley IFS

˜ ˜ ˜ ˜ ˜ N ˜ bi 0 ˜ ti,1 = fi(x) := Aix + ti i=1, where Ai = and ti = , F { } 0 a˜i t˜i ,2 e and its attractor Λ is the GL brother of and Λ, respectively, if a˜ = a and b˜ = b for F i i i i every i [N], furthermore, has the same column structure (2.1.10) as . If the shifted ∈ F F TGL carpet Λ is actuallye a TGL (that is Λ has non-overlapping column structure) then we also require that ti,1 = ti,1 holdse for all i [N]. ∈ M There always exists such a brother since we assume Deﬁnition 2.1.1 (c) and ∑ rıˆ 1. ıˆ=1 ≤ Throughout, thee GL brother of Λ will always be denoted with the extra tilde Λ.

e 2.1. Triangular Gatzouras–Lalley-type carpets 21

Some notation The map f is indexed by i [N]. To indicate which column i belongs to in the i ∈ partition (2.1.4) we use the function

φ : 1, 2, . . . , N 1, 2, . . . , M , φ(i) := ıˆ if i . (2.1.9) { } → { } ∈ Iıˆ With this notation we can formulate the column structure (2.1.5) as

if φ(k) = φ(`) = ıˆ, then bk = b` =: rıˆ and tk,1 = t`,1 =: uıˆ. (2.1.10)

Throughout, i is an index from [N], while ıˆ with the hat is an index corresponding to a column from 1, . . . , M . We use analogous notation for inﬁnite sequences i = i i ... { } 1 2 and ıˆ = ıˆ1ıˆ2 . . ., see Subsection 2.3.1 for details. For compositions of maps we use the standard notation f := f f ... f , i1...in i1 ◦ i2 ◦ ◦ in where i 1, . . . , N . Similarly, for products of matrices we write j ∈ { }

bi1...in 0 Ai1...in := Ai1 ... Ain := . · · di i ai i 1... n 1... n Immediate calculations give b = b ... b , a = a ... a and i1...in i1 · · in i1...in i1 · · in n n d = d a b i1...in ∑ i` ∏ ik ∏ ir , (2.1.11) `=1 · k<` · r=`+1

n where by deﬁnition ∏ aik := 1 and ∏ bir := 1. The image Ri1...in := fi1...in (R) is a k<1 r=n+1 parallelogram with two vertical sides, see Figure 2.1. We refer to bi1...in as the width,

ai1...in as the height and γi1...in as the angle of the longer side of the parallelogram Ri1...in , in other words di1...in tan γi1...in := . (2.1.12) bi1...in n i ... 1 i a

in |

... n R i 1 i ... 1 i d | γi1...in

bi1...in

2 FIGURE 2.1: The skewness of Ri1...in := fi1...in ([0, 1] )

Since direction-x dominates, Ri1...in is extremely long and thin for large n. A simple argument gives that tan γ remains uniformly bounded away from +∞. | i1...in | 22 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

Lemma 2.1.5. There exists a non-negative constant K0 < ∞ such that for every n and every ﬁnite length word i1 ... in d i1...in K . b ≤ 0 i1...in

Proof. Since direction-x dominates, max a /b < 1, hence using (2.1.11) i{ i i} n k 1 d d di − aij max d /b i1...in | i1 | + | k | i{| i| i} < ∞. b ≤ b ∑ b ∏ b ≤ 1 max a /b i1...in i1 k=2 ik j=1 ij i i i − { }

2.1.1 Results of Gatzouras and Lalley A standard technique to give a lower bound for the Hausdorff dimension of the at- ˜ tractor Λ = i [N] fi(Λ) is to study self-afﬁne measures νp, i.e. compactly supported ∈ measures withS support Λ satisfying e e N e ˜ 1 νp = ∑ piνp fi− , i=1 ◦ for some probability vector p = (p ,..., p ). Let be the set of all probability distri- 1 N P butions on the set [N] and be the subset when all p > 0. By deﬁnition P0 i

sup dimH νp dimH Λ. p ≤ ∈P e Gatzouras and Lalley proved that there always exists a p∗ for which the supremum is attained, furthermore p . Let ∗ ∈ P0

α∗ := dimH νp∗ = sup dimH νp. p ∈P They explicitly calculated

N N M ∑i=1 pi log pi ∑i=1 pi log bi ∑ıˆ=1 qıˆ log qıˆ dimH νp = N + 1 N N , (2.1.13) ∑i=1 pi log ai − ∑i=1 pi log ai ! ∑i=1 pi log bi

where qıˆ = ∑j ıˆ pj. This formula is a special case of the Ledrappier–Young formula, see Subsection∈I 2.3.3 for details and references. For Bedford–McMullen carpets the optimal p∗ can be given by routine use of the Lagrange multipliers method. The main result of [GL92] is that for a GL carpet the α∗ bound is sharp, i.e.

α∗ = dimH Λ.

In [GL92] Gatzouras and Lalley also gave ane implicit formula to calculate the box M sx dimension of their carpet. Let sx be the unique real such that ∑ıˆ=1 rıˆ = 1 (rıˆ was deﬁned in (2.1.5)). Then dimB Λ = s is the unique real such that

N e sx s sx ∑ bi ai − = 1. i=1 2.1. Triangular Gatzouras–Lalley-type carpets 23

Again, equality of dimH Λ and dimB Λ is highly atypical. It holds if and only if the α∗- dimensional Hausdorff measure of Λ, denoted α∗ (Λ), is positive and ﬁnite, which H is equivalent to the conditione e e e α∗ sx ∑ aj − = 1, for every ıˆ = 1, . . . , M. j ∈Iıˆ

For Bedford–McMullen carpets Peres showed in [Per94] that α∗ (Λ) = ∞ when H dimH Λ < dimB Λ. e 2.1.2e Separatione conditions In our main results, we assume different extents of separation for the parallelograms fi(R), recall Figures 1.6 and 1.9. This will be considered in Subsection 2.1.2. In Sub- section 2.1.2 we consider separation conditions for which are actually conditions H about the extent of separation of the column structure.

Separation of the cylinder parallelograms Definition 2.1.6 (Separation conditions for a shifted TGL IFS ). We say that F satisfies the rectangular open set condition (ROSC): recall Definition 1.2.1. • F each column independently satisfies the ROSC if for every ıˆ [M] and • ∈ k, ` we have f (U) f (U) = ∅. In other words, if the interior of two first level ∈ Iıˆ k ∩ ` cylinders intersects, then they are from different columns.

satisfies the transversality condition if there exists a K1 > 0 such that for • F n every n and words (i ... i ), (j ... j ) 1, . . . , N with φ(i ) = φ(j ) for k = 1 n 1 n ∈ { } k k 1, . . . , n and i = j (φ was defined in (2.1.9)), we have 1 6 1 proj (int(R ) int(R )) < K max a ... a , a ... a . (2.1.14) | x i1...in ∩ j1...jn | 1 · { i1 · · in j1 · · jn } Given two finite words i ... i and j ... j , i = j , the angle of the two corre- 1 n 1 n 1 6 1 sponding parallelograms Ri1...in and Rj1...jn can be defined as the angle between their non-vertical sides. The transversality condition ensures that any such pair of parallelograms in the same column have either disjoint interior or have an angle uniformly separated from zero. Observe that this definition of transversality coincides in the diagonally homogeneous case with the one in [BRS]. In [BRS, Section 1.5] a sufficient condition for the transversality condition was given. Namely, the authors introduced a self-affine IFS in R3 which is (in our setup) S N b 2 := Si(x, z) := ( fi(x), Ti(z)) , (x, z) [0, 1] R, S i=1 ∈ × n o where f N bwas definedb in (2.1.1) and T : R R is given by { i}i=1 i → a b N := T (z) := i z + i . T i b · d i i i=1 The relevance of the IFS is that T 24 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

ξ ) (` A i

Ti(z) T

` z

(1, 0)

ai di Ti(z) := z + bi · bi

FIGURE 2.2: The IFS , where z and (1, z) are identiﬁed T

tan γi1...in = Ti1...in (0). (2.1.15)

Indeed, from the deﬁnition (2.1.12) of tan γi1...in and formula (2.1.11) it immediately follows that n ` 1 di1 di` − aik tan γi1...in = + ∑ ∏ = Ti1...in (0). bi1 `=2 bi` · k=1 bik Using the same argument as in the proof of [BRS, Lemma 1.2] we obtain that

Lemma 2.1.7. If satisﬁes the Strong Separation Property (that is S (Λ) S (Λ) = ∅ if S i ∩ j i = j and Λ is the attractor of the IFS ) then the transversality condition holds. 6 b S b b b b The nextb lemma gives a different,b easy-to-check sufﬁcient condition for transversality.

Lemma 2.1.8. Let := (k, `) : k, ` , k = `, R R = ∅ , where A denotes Pˆ ∈ Iˆ 6 k◦ ∩ `◦ 6 ◦ the interior of a set A. Moreover, we introduce

dk ak sk := , rk := , r∗ := max rk, bmin := min bk and s := min min sk s` . 1 ˆ M bk bk 1 k N 1 k N ∗ (k,`) ˆ | − | ≤ ≤ ≤ ≤ ≤ =≤∅ Pˆ6 ∈P Assume that 1 r∗ s bmin s > 2 or equivalently ∗ > r∗. ∗ bmin · 1 r∗ 2 + s bmin − ∗ Then the transversality condition holds. In particular, in the diagonally homogeneous case transversality holds if

a d < ∗ , (2.1.16) b 2 + d ∗ where d := min min dk d` . 1 ˆ M ∗ (k,`) ˆ | − | ≤ =≤∅ Pˆ6 ∈P Proof. Using that R [0, 1]2 we obtain that d < 1. Hence k ⊂ | k| 1 sk . (2.1.17) | | ≤ bmin 2.1. Triangular Gatzouras–Lalley-type carpets 25

For an m 1, . . . , M let Σ := j Σ, j . The transversality condition holds ∈ { } m { ∈ 1 ∈ Im} if there exists c > 0 such that for every n, for all m M with = ∅ and ≤ Pm 6

for all i, j Σ with (i1, j1) m, we have: γi n γj n > c. (2.1.18) ∈ ∈ P | − |

It follows from (2.1.15) that (2.1.18) holds whenever for all such pair of i, j and for all n n ` 1 ` 1 s s s − r s − r i1 j1 ∑ i` ∏ ik j` ∏ jk | − | − `=2 · k=1 − · k=1 ! is greater than the same positive constant uniformly. However by (2.1.17) this holds if 1 r s > 2 ∗ . ∗ b · 1 r min − ∗

Separation of the columns We will also need some separation conditions for the column structure which are represented by separation properties of , recall (2.1.6). H The symbolic spaces for and are F H Σ := 1, . . . , N N and Σ := 1, . . . , M N . { } H { } The natural projections form Σ Λ and Σ Λ are Π and Π respectively, → H → H H see Subsection 2.3.1 for details. Whenever we are given a probability vector p on 1, . . . , N , we always associate to it another probability vector q on 1, . . . , M such { } { } that qıˆ := ∑ pj. (2.1.19) j ∈Iıˆ Slightly abusing the notation we write for both the set of the probability vectors P0 of positive components on 1, . . . , N and 1, . . . , M . The Bernoulli measure pN on { } { } 1 Σ is denoted µp and its push forward is νp = Π µp = µp Π− . Analogously for µq ∗ ◦ and νq.

Deﬁnition 2.1.9 (Separation conditions for ). We say that satisﬁes H H Hochman’s Exponential Separation Condition (HESC) (see [Hoc14, p. 775]) if • there exist an ε > 0 and n ∞ such that for k ↑

hı(0) h(0) , if hı0(0) = h0(0); ∆n := min | − | ı, 1...M n ∞, otherwise. ∈{ı= } 6

ε nk we have ∆nk > e− · . Here h0 denotes the derivative of the function h. Weak Almost Uniqe Coding (WAUC) if for all Bernoulli measures µ there exists • q Σ (may depend on q) for which BH ⊂ H 1 µq( ) = 0 and for every ıˆ Σ :#(Π− Π (ıˆ) ) = 1. BH ∈ H \BH H H \BH

Almost Uniqe Coding (AUC) holds if for every Bernoulli measure µq and for µq-a.e. 1 ıˆ Σ :#Π− Π (ıˆ) = 1. ∈ H H H 26 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

No Dimension Drop (NDD) if for all push forward measures νq = (Π ) µq • H ∗ M ∑ıˆ=1 qıˆ log qıˆ dimH νq = − M . ∑ qıˆ log rıˆ − ıˆ=1 The following implications hold between these conditions

HESC = NDD WAUC. (2.1.20) ⇒ ⇐⇒ HESC = NDD follows from Hochman’s work [Hoc14, Theorem 1.1]. AUC im- ⇒ plies NDD from Feng–Hu [FH09, Theorem 2.8 and Corollary 4.16], but we do not know if the reverse direction NDD = AUC holds or not. Feng informed us [Fen19, ⇒ Corollary 4.7] that he can prove the equivalence NDD WAUC for ergodic mea- ⇐⇒ sures. This result just appeared on the arXiv. However, we use it only for Bernoulli measures. For completeness, we give our own complete (much simpler) proof of NDD WAUC for Bernoulli measures in AppendixB. ⇐⇒ The set of translations (u ,..., u ) deﬁning for which HESC does not hold U 1 M H is small. It is stated in [PS, Proposition 2.7] that it essentially follows from [Hoc15, Theorem 1.10] that the Hausdorff and packing dimension of is M 1, in particular U − has 0 M-dimensional Lebesgue measure. Moreover, [Hoc14, Theorem 1.5] states U that if the parameters (r ,..., r , u ,..., u ) deﬁning are all algebraic, then HESC 1 M 1 M H does not hold if and only if there is an exact overlap, i.e. ∆n = 0 for some n.

2.2 Main results

We now state our main results for the Hausdorff dimension of shifted TGL carpets in Subsection 2.2.1, the box dimension in Subsection 2.2.2 and discuss diagonally homogeneous carpets in Subsection 2.2.3. For a discussion on generalizing towards negative entries in the main diagonal, see Subsection 2.7.5.

2.2.1 Hausdorff dimension

For any vector c = (c1,..., cK) with strictly positive entries and a probability vector p = (p1,..., pK) we write K pi c p := ∏ ci . h i i=1 When no confusion is made, we suppress p and write c = c . Throughout, we h i h ip use this notation for the vectors a = (a1,..., aN), b = (b1,..., bN), p = (p1,..., pN), N = (N1,..., NM) and q = (q1,..., qM), where q is derived from p via (2.1.19). Using this notation let us denote the function on the right-hand side of (2.1.13) by

log p p log b p log q q log q log p log q D(p) := h i + 1 h i h i = h i + h i − h i. (2.2.1) log a − log a log b log b log a h ip h ip h ip h i h i Theorem 2.2.1 (Upper bound). Regardless of overlaps, for any shifted triangular Gatzouras– Lalley-type planar carpet Λ

dimH Λ sup D(p) =: α∗. ≤ p ∈P Furthermore, there always exists a p for which D(p ) = α . ∗ ∈ P0 ∗ ∗ 2.2. Main results 27

The proof is given in Section 2.4. Throughout, let q∗ denote the vector qı∗ˆ =

∑j ıˆ p∗j . The next theorem states sufﬁcient conditions under which the Hausdorff ∈I dimension of a self-afﬁne measure νp on Λ is equal to D(p).

N Theorem 2.2.2. Let p 0, µp := p and νp := Π µp. For a shifted triangular ∈ P ∗ Gatzouras–Lalley-type planar carpet Λ we have

dimH νp = D(p) if the horizontal IFS satisﬁes Hochman’s Exponential Separation Condition (in particular, H always holds for non-overlapping columns) and

(i) either each column independently satisﬁes the ROSC or

(ii) Λ satisﬁes transversality (see Deﬁnition 2.1.6) and the following inequality holds:

log a p log N q h i > 1 + h i . (2.2.2) log b log q h ip − h iq We remark that Proposition 2.2.10 provides a simple way to check condition (2.2.2) in the diagonally homogeneous case. Section 2.5 is devoted to the proof of this theorem. As an immediate corollary, we get

Corollary 2.2.3 (Sufﬁcient conditions). Whenever a shifted TGL carpet Λ satisﬁes the conditions of Theorem 2.2.2 with replacing p and q in (2.2.2) by p∗ and q∗, then

dimH Λ = α∗.

2.2.2 Box dimension

Recall the IFSs (2.1.2) and (2.1.6) obtained by projecting to the x-axis. Recall sx H M sx H F N s˜x was defined so that ∑ıˆ=1 rıˆ = 1 and let s˜x be the unique real such that ∑i=1 bi = 1. Furthermore, introducee s := dimB Λ = dimB Λ . H H H Since Λ is a self-similar set, s is well defined. If Λ is a TGL carpet then s = sx, H H e H otherwise s sx. The affinity dimension dimAff of Λ can be deduced from the H ≤ result of Falconer–Miao [FM07, Corollary 2.6] together with the description in [BRS, Subsection 1.3] and the fact that direction-x dominates: dimAff Λ = sA is the unique real such that N min s˜x,1 sA min s˜x,1 ∑ bi { }ai − { } = 1. (2.2.3) i=1

N sA 1 In particular, if s˜x < 1 then sA = s˜x, otherwise sA solves ∑i=1 biai − = 1. So sA only depends on the main diagonals (bi, ai), but not on the off-diagonal elements di. So, the afﬁnity dimension of a shifted TGL carpet Λ is and its GL brother coincide. The following theorem gives an upper bound for dimB Λ, which can be strictly smaller than sA. It was proved for diagonal iterated function systems by Feng–Wang in [FW05, Corollary 1] and also follows from Fraser’s work [Fra12, Theorem 2.4, Corollary 2.7]. Here we extend its scope to triangular IFSs. In a different context, Hu [Hu98] studied a related problem, where a version of Bowen’s formula determines the box dimension. 28 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

Theorem 2.2.4 (Upper bound). Regardless of overlaps, for any shifted triangular Gatzouras– Lalley-type planar carpet Λ

dim Λ = dim Λ s s , P B ≤ ≤ A where s is the unique solution of the equation

N s s s ∑ bi H ai − H = 1. (2.2.4) i=1

In particular, if Λ satisﬁes the ROSC, then dimP Λ = dimB Λ = s. Corollary 2.2.5 (Equality of box- and afﬁnity dimension). For any shifted TGL carpet

s = sA s = min s˜x, 1 . ⇐⇒ H { }

Proof. Follows immediately from comparing equations (2.2.3) and (2.2.4) defining sA and s, respectively, together with the fact that ai < 1 and bi/ai > 1 for every i = 1. . . . , N. Remark 2.2.6. a) The proof of Fraser [Fra12] does not make use of any column structure (2.1.10). Hence, Theorem 2.2.4 immediately extends to an IFS of the form (2.1.1) as long as direction-x F dominates (0 < ai < bi < 1) and the ROSC holds. b) Since Λ is compact and every open set intersecting Λ contains a bi-Lipschitz image of Λ, we get that dimP Λ = dimBΛ, see [Fal90, Corollary 3.9]. Handling overlaps to calculate the box dimension is a greater challenge, since typically dimH Λ < dimB Λ and thus the usual technique of giving a lower bound by bounding the Hausdorff dimension from below does not suffice. Hence, a new counting argument was necessary. Theorem 2.2.7 (Box dimension with overlaps). For a shifted TGL carpet Λ we have dim Λ s, hence dim Λ = s, if either of the following hold: B ≥ B (i) satisfies HESC and each column independently satisfies the ROSC or H (ii) Λ is a TGL carpet, satisfies transversality and the following inequality:

log p p + log q q < s (log b p log a p), (2.2.5) − h i h i H h i − h i where p := (p1,..., pN)eande q := (eq1,...,e qM) are definede by equatione (2.2.4): s s s s s s e e e pi = bei H ai − He and eqıˆ = ∑ bj H aj− H . (2.2.6) j ∈Iıˆ e e The analogue of the following sufficient and necessary condition for the equality of the box- and Hausdorff dimensions was proved in [GL92, Theorem 4.6]. Theorem 2.2.8 (Equality of box- and Hausdorff dimension). Assume the shifted TGL carpet Λ satisfies ROSC and satisfies No Dimension Drop. Then the following three con- H ditions are equivalent,

s s dimH Λ = dimB Λ s = dimH νq ∑ aj− H = 1 for every ıˆ [M]. (2.2.7) ⇐⇒ H ⇐⇒ j ∈ ∈Iıˆ e All results for box dimension are proved in Section 2.6. 2.2. Main results 29

2.2.3 Diagonally homogeneous carpets We show how the conditions and formulas of our main results simplify in the diagonally homogeneous case. Recall the easy-to-check sufﬁcient condition (2.1.16) for transversality in Lemma 2.1.8. Moreover, observe that the vector p becomes the uniform vector pi = 1/N and thus qıˆ = Nıˆ/N. A routine use of the Lagrange multipliers method gives the optimal p∗ e e e log b 1 M log b 1 log a log a − p∗ = N − N if k . (2.2.8) k ıˆ · ∑ ˆ ∈ Iıˆ ˆ=1 Thus, conditions (2.2.2) and (2.2.5) become

log p∗ p log a log N log q q log a h i ∗ < and + 1 + h i < , (2.2.9) log q∗ q log b log M log M log b h i ∗ e e respectively. If in addition, the system has uniform vertical ﬁbres, then pi = pi∗ = 1/N also qıˆ = qı∗ˆ = 1/M. Hence, both conditions (2.2.2) and (2.2.5) become e log N log a e < . (2.2.10) log M log b

Next, we give an equivalent explicit formulation of condition (2.2.2). Let ϕ(y) := y log y and for x (0, 1) deﬁne ∈ M x 1 ϕ ∑ıˆ=1 Nıˆ R(x) := x + (r(x) 1)− , where r(x) = . M x − ∑ˆ=1 ϕ(Nˆ )

Lemma 2.2.9. R(x) is a continuous, strictly monotone increasing function.

Proof. Continuity is obvious. It is enough to show that r(x) is strictly monotone decreasing. Let r0 denote the derivative. Then

=:A =:B

M 2 M 2 M M 2 x x x x x ϕ(N ) r0(x) = ϕ(N ) + log N ϕ(N ) · ∑ ˆ · z ∑ }| ˆ { z ∑ ıˆ }|· ∑ ˆ { ˆ=1 ˆ=1 ıˆ=1 ˆ=1 >0 M M M M | {z } ϕ Nx ϕ(Nx) ϕ Nx ϕ(Nx) log Nx . − ∑ ıˆ · ∑ ˆ − ∑ ıˆ ∑ ˆ · ˆ ıˆ=1 ˆ=1 ıˆ=1 ˆ=1 =:C =:D

We claim that C > A and D| B, which{z will conclude} | the proof{z of the lemma.} For ≥ brevity, write y := Nx. y = 1 N = 1, otherwise y > 1. ıˆ ıˆ ıˆ ⇔ ıˆ ıˆ To show that C > A, it is enough to prove that for 1 u v ≤ ≤ ϕ(u) + ϕ(v) < ϕ(u + v). (2.2.11) 30 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

Then a simple induction implies that ∑ ϕ(yıˆ) < ϕ(∑ yıˆ). Recall ϕ(1) = 0. The mean value theorem implies that

ϕ(u + v) ϕ(v) = u ϕ0(ξ), for some ξ (v, u + v) − · ∈ ϕ(u) ϕ(1) = (u 1) ϕ0(ζ), for some ζ (1, u). − − · ∈

Since the derivative ϕ0(y) = 1 + log y is strictly increasing and ζ < ξ, we have ϕ0(ζ) < ϕ0(ξ). This implies (2.2.11). To prove the other inequality

M M M M M 2 D B = y log y ϕ(y ) log y log y ϕ(y ) . − ∑ ıˆ · ∑ ıˆ ∑ ˆ · ˆ − ∑ ıˆ · ∑ ˆ ıˆ=1 ıˆ=1 ˆ=1 ıˆ=1 ˆ=1

We can pull out log ∑ yıˆ > 0 and divide by it. This gives

M M M D B yˆ log yˆ 2 − = ∑ yıˆ ∑ ϕ(yˆ) ∑ ϕ (yˆ) 2 ∑ ϕ(yıˆ)ϕ(yˆ) log ∑ yıˆ · yˆ − − ıˆ=1 ˆ=1 ˆ=1 ıˆ<ˆ M M M ∑ıˆ=1 yıˆ 2 2 = ∑ ϕ (yˆ) ∑ ϕ (yˆ) 2 ∑ ϕ(yıˆ)ϕ(yˆ) ˆ=1 yˆ · − ˆ=1 − ıˆ<ˆ M ∑i=j yıˆ 2 = ∑ 6 ϕ (yˆ) 2 ∑ ϕ(yıˆ)ϕ(yˆ) ˆ=1 yˆ · − ıˆ<ˆ

yıˆ 2 yˆ 2 = ∑ ϕ (yˆ) + ϕ (yıˆ) 2ϕ(yıˆ)ϕ(yˆ) yˆ · yıˆ · − ıˆ<ˆ 2 yıˆ yˆ = ∑ ϕ(yˆ) ϕ(yıˆ) 0. yˆ − yıˆ ≥ ıˆ<ˆ r r

Proposition 2.2.10. The solution of the equation R(x) = 1 is unique. Let x0 denote this solution. Then in the diagonally homogeneous case

log b (2.2.2) holds < x . ⇐⇒ log a 0

Remark 2.2.11. Observe that all the conditions for transversality, (2.2.2), (2.2.5) are satisﬁed if the heights of the parallelograms Ri are "small enough" compared to their width. See the examples with overlaps in Section 2.7 for some explicit calculations.

Proof of Proposition 2.2.10. Let x := log b/ log a < 1. In the diagonally homogeneous case (2.2.2) simpliﬁes to

M log a 1 ∑ q log Nıˆ = > 1 + ıˆ=1 ı∗ˆ , log b x ∑M q log q − ıˆ=1 ı∗ˆ ı∗ˆ x x where qı∗ˆ = Nıˆ / ∑ˆ Nˆ . Multiplying each side by x we get

∑M q log Nx 1 > x + ıˆ=1 ı∗ˆ ıˆ . (2.2.12) ∑M q log q − ıˆ=1 ı∗ˆ ı∗ˆ 2.3. Preliminaries 31

y yˆ It is straightforward to check that for any y ,..., yM R and qıˆ := e ıˆ / ∑ e 1 ∈ ˆ M M M yıˆ ∑ qıˆ log qıˆ + ∑ qıˆ yıˆ = log ∑ e . − ıˆ=1 ıˆ=1 · ıˆ=1

x Applying this with yıˆ = log Nıˆ (then qıˆ = qı∗ˆ ) in the denominator of (2.2.12) we get that (2.2.2) is equivalent to

∑M q log Nx 1 > x + ıˆ=1 ı∗ˆ ıˆ = R(x). log ∑M Nx ∑M q log Nx ıˆ=1 ıˆ − ıˆ=1 ı∗ˆ ıˆ For x small enough (2.2.2) holds, since 1/x tends to infinity while the right hand side remains finite. On the other hand for x = 1 it does not hold. Hence, R(x) < 1 for small enough x, while R(1) 1. Thus, Lemma 2.2.9 implies that there exists a unique ≥ x (0, 1) such that R(x ) = 1. So any x < x satisfies (2.2.2). 0 ∈ 0 0 Finally, in the diagonally homogeneous case, the dimension formulas agree with the ones for Bedford–McMullen carpets. Corollary 2.2.12. If a diagonally homogeneous shifted TGL carpet Λ satisfies the conditions of Theorems 2.2.2 and 2.2.7, then

1 M log b log N log b log M dim Λ = log N log a and dim Λ = + 1 . H log b ∑ ˆ B log a − log a log b − ˆ=1 − −

In particular, dimH Λ = dimB Λ if and only if Λ has uniform vertical ﬁbres. Proof. For diagonally homogeneous shifted TGL carpets the expression (2.2.1) for D(p) simpliﬁes to

log p p log b log q q D(p) = h i + 1 h i . log a − log a log b

Applying this for p∗ from (2.2.8) gives the result dimH Λ = D(p∗). The equation for the box dimension s = dimB Λ, recall (2.2.4), simpliﬁes to

s s s N b H a − H = 1. (2.2.13) · · Since has No Dimension Drop (recall Deﬁnition 2.1.9), we have s = log M/( log b). H H − Substituting this back into (2.2.13) and expressing s from the equation gives the de- sired formula for dimB Λ. Comparing the formula for dimB Λ with the one for D(p), we immediately get that equality holds if and only if N = N/M for every ıˆ 1, . . . , M . ıˆ ∈ { } 2.3 Preliminaries

In this section, we collect important notation, deﬁnitions, preliminary lemmas and cite results used in the proofs of the subsequent sections.

2.3.1 Symbolic notation

Throughout, we work simultaneously with the IFSs , and , which were deﬁned F H H in (2.1.1), (2.1.2) and (2.1.6) respectively. Their attractors are Λ, Λ = Λ respectively. e H H e 32 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

We deﬁne the symbolic spaces

Σ = 1, 2, . . . , N N and Σ = 1, 2, . . . , M N { } H { } with elements i = i1i2 ... Σ and ıˆ = ıˆ1ıˆ2 ... Σ . The function φ : 1, 2, . . . , N ∈ ∈ H { } → 1, 2, . . . , M , recall (2.1.9), naturally deﬁnes the map Φ : Σ Σ { } → H

Φ(i) := ıˆ = φ(i1)φ(i2) . . . . (2.3.1)

Finite words of length n are either denoted with a ’bar’ like ı = i ... i Σ or as 1 n ∈ n a truncation i n = i ... i of an infinite word i, the length is denoted . The set of | 1 n | · | all finite length words is denoted by Σ∗ = n Σn and analogously Σ∗ . The left shift (i) = i i (ıˆ) = ıˆ ıˆ H operator on Σ and Σ is σ, i.e. σ 2 3 . .S . and σ 2 3 .... The longest commonH prefix of i and j is denoted i j, i.e. its length is i j = ∧ | ∧ | min k : i = j 1. This is also valid if one of them has or both have finite length. { k 6 k} − The nth level cylinder set of i Σ is [i n] := j Σ : i j n . Similarly for ı Σn ∈ 2 | { ∈ | ∧ | ≥ } ∈ and ıˆ Σ . Recall that R = [0, 1] . We use the standard notation Ai n = Ai1 ... Ain ∈ H | · · and fi n = fi1 fi2 ... fin to write | ◦ ◦ ◦

Λi n := fi n(Λ) and Ri n := fi n(R) | | | | ∞ for the nth level cylinder corresponding to i. The sets Ri n n=1 form a nested se- { | } quence of compact sets with diameter tending to zero, hence their intersection is a unique point x Λ. This deﬁnes the natural projection Π : Σ Λ ∈ → ∞ ∞

Π(i) := lim Ri n = lim fi n(0) = ti + Ai n 1 tin . (2.3.2) n ∞ n ∞ 1 ∑ | | n= | − · → n\=1 → 2 The natural projections generated by and are H H

Π (i) := lim hi n(0), i Σ;e and Π (ıˆ) := lim hıˆ n(0), ıˆ Σ . H n ∞ H n ∞ H → | ∈ → | ∈ The followinge commutativee diagram summarizes these notations:

Φ Σ / Σ (2.3.3) H Π Π H Π H e Λ / Λ projx H

We also introduce the measurable partitions α and β of Σ whose classes containing an i Σ are deﬁned ∈ 1 1 α(i) := Π− Π(i) and β(i) := Φ− Φ(i). (2.3.4) The fact that these partitions are measurable are immediate consequences of the definition of measurability of a partition. Alternatively, this also follows from [Sim12, Theorem 2.2]. Thus, α(i) contains those j Σ which get mapped to the same point ∈ on the attractor, Π(i) = Π(j) Λ, and β(i) corresponds to the ’symbolic column’ of ∈ i, i.e. for j β(i) we have Π (i) = Π (j). These partitions play an important role ∈ H H when handling overlaps. e e 2.3. Preliminaries 33

Bernoulli measures on Σ are key in obtaining the lower bound for dimH Λ. Recall the set N := p = (p1,..., pN) : pi 0, ∑ pi = 1 P { ≥ i=1 } of all probability distributions on the set 1, 2, . . . , N and let denote the subset { } P0 when all p > 0. The Bernoulli measure on Σ corresponding to p is the product i ∈ P measure µ = pN, i.e. the measure of a cylinder set is µ ([i n]) = p ... p . All p p | i1 · · in Bernoulli measures can be uniquely disintegrated according to the family of conditional measures µp,α(i) = µα(i) generated by the measurable partition α. That is for all Borel sets U Σ ⊂ µp(U) = µα(i)(U)dµp(i). (2.3.5) Z The entropy of a Bernoulli measure µp is

N hµp = ∑ pi log pi = log p p. (2.3.6) − i=1 − h i

The push forward νp := Π µp is the self-afﬁne measure on Λ deﬁned by νp = µp 1 ∗ ◦ Π− or equivalently N 1 νp = ∑ piνp fi− . i=1 ◦

Recall that a p deﬁnes another distribution q = (q1,..., qM) via (2.1.19). Then N ∈ P µq = q is a Bernoulli measure on Σ . Moreover, the self-similar measure on Λ is H H νq = (Π ) µp = (proj ) νp. Our convention is that µ always denotes a measure on ∗ x ∗ (some) symbolicH space, while ν is supported on (a part of) R.

2.3.2 Atypical parallelograms The exponential rate of growth of the size of nth level parallelograms, the number of parallelograms in a column and the column’s measure can vary a lot for different i Σ. However, in measure-theoretic sense those i which behave atypically form a ∈ small set. Deﬁne the function

X : Σ R+, X(i) := c , → i1 where c = (c1,..., cN) is an arbitrary vector with strictly positive elements. Let

n 1 n − j Xn(i) := ∏ X(σ i) = ∏ cij . j=0 j=1

In particular, if c = a := (a1,..., aN) or b := (b1,..., bN), then Xn(i) is the height and width of the parallelogram Ri n. If c = N := (Nφ(1),..., Nφ(N)) or q := (qφ(1),..., qφ(N)), | then Xn(i) gives the number of parallelograms in and the measure of the column Φ(i) n. | N pj Fix an arbitrary p . Recall the notation c p := ∏ c . When no confusion is ∈ P h i j=1 j made, we suppress p and write c = c . In the rest of the subsection δ > 0 is ﬁxed. h i h ip Deﬁne

i Σ : X (i) < c (1 δ)n or X (i) > c (1+δ)n , if c > 1, (c) = n − n Badδ,n : { ∈ h i(1+δ)n h i(1 δ)n} h i (2.3.7) ( i Σ : Xn(i) < c or Xn(i) > c − , if c < 1. { ∈ h i h i } h i 34 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

The deﬁnition can be extended to a positive real t, by setting Badδ,t(c) := Badδ, t (c). Let µ be the Bernoulli measure on Σ deﬁned by p . b c p ∈ P Lemma 2.3.1. IF c = 1 then there exists a constant C and an r (0, 1) such that h ip 6 ∈ µ (Bad (c)) < C rn for every n 1. p δ,n · ≥ Hence, the Borel-Cantelli lemma immediately implies that

µ (i Bad (c) for inﬁnitely many n) = 0. p ∈ δ,n 1 n 1 j Proof. Assume c > 1. Let Sn(X) := ∑ − log X(σ i). Then h i n j=0 (1 δ)n µ (X (i) < c − ) = µ (S (X) < (1 δ) log c ). p n h i p n − h i The log X(σji) are independent and identically distributed with expectation { }j N E (log X) = ∑ pj log cj = log c . j=1 h i

Hence, Cramér’s large deviation theorem [DZ10, Theorem 2.1.24.] implies that µp(Xn(i) < c (1 δ)n) decays exponentially fast in n. The argument for X (i) > c (1+δ)n is ex- h i − n h i actly the same. The proof is analogous when c < 1. h i 2.3.3 Ledrappier–Young formula Let 0 < α (A) α (A) < 1 denote the two singular values of a 2 2 contractive, 2 ≤ 1 × non-singular matrix A. Namely, αi(A) is the positive square root of the ith largest eigenvalue of AT A, where AT is the transpose of A. The geometric interpretation of the singular values is that the linear map x Ax maps the unit disk to an ellipse 7→ with principal semi-axes of length α2(A) and α1(A). The singular values can also be expressed with the matrix norm: α (A) = A and α (A) = A 1 1. For a family 1 k k 2 k − k− of matrices = A ,..., A , the asymptotic exponential growth rate of the semi- A { 1 N} axes of the ellipses determined by the maps x A x is given by the Oseledets 7→ i1...in theorem. Theorem 2.3.2 (Oseledets [Ose68]). Let = A ,..., A be a set of non-singular 2 2 A { 1 N} × matrices with A < 1 for i 1, . . . , N . Then for any ergodic σ-invariant measure µ on k ik ∈ { } Σ there exist constants 0 < χ1 χ2 such that for µ-almost every i µ ≤ µ

1 1 1 lim log α1(Ai ...i ) = lim log Ai ...i = χ , n ∞ n 1 n n ∞ n 1 n µ → → k k − 1 1 1 1 2 lim log α2(Ai ...i ) = lim log (Ai ...i )− − = χ . n ∞ n 1 n n ∞ n 1 n µ → → k k − 1 2 1 2 The numbers χµ and χµ are called the Lyapunov-exponents of ν = Π µ. If χµ = χµ then we ∗ 6 say that µ has simple Lyapunov spectrum. It is an easy exercise to calculate the Lyapunov exponents of Bernoulli measures µp for a family of lower triangular matrices for which direction-x dominates. For greater generality see Falconer–Miao [FM07]. Lemma 2.3.3. Fix any p and a family of lower triangular matrices = A ,..., A ∈ P A { 1 N} for which direction-x dominates. Then the Lyapunov spectrum of the Bernoulli measure µp is 2.4. Upper bound for dimH Λ 35 simple and the exponents equal

N N χ1 = p b = b and χ2 = p a = a νp ∑ i log i log p νp ∑ i log i log p. − i=1 − h i − i=1 − h i

Sketch of proof. Both the singular values or the norm of Ai1...in can be calculated directly. Since direction-x dominates, the off-diagonal element does not play a role. An application of Oseledets theorem and the strong law of large numbers concludes the proof.

The Ledrappier–Young formula originates from the seminal work of Ledrappier and Young [LY85a; LY85b] on determining the Hausdorff dimension of invariant measures of diffeomorphisms on compact manifolds. Through a succession of papers by Przytycki–Urbański[PU89], Feng–Hu [FH09], Bárány [B15´ ] and Bárány–Käenmäki [BK17] the formula was proved for the Hausdorff dimension of wider and wider classes of self-affine measures. In fact, Feng [Fen19] recently announced that the Hausdorff dimension of the push-forward of a shift-invariant, ergodic measure µ satisfies a Ledrappier–Young type formula in full generality for any self-affine IFS on Rd which is contracting on average with respect to µ. Also observe that the formulas proved in the earlier works of [Bar07; Bar08; Bed84; GL92; McM84] are all special cases of the Ledrappier–Young formula. The main result of [BK17, Theorem 2.4, Corollary 2.8] can be stated in a simpler form in our context when direction-x dominates. Theorem 2.3.4 ([BK17], direction-x dominates). Let be a shifted TGL-type IFS of the F form (2.1.1). Furthermore, using the notation from Subsection 2.3.1, let µp be any Bernoulli measure on Σ, νp = Π µp its push forward and νq = (proj ) νp. Then, regardless of ∗ x ∗ overlaps, νp is exact dimensional and satisfies the Ledrappier–Young formula

h H χ1 µp − νp dimH νp = 2 + 1 2 dimH νq, (2.3.8) χνp − χνp ! where H = log µ ([i ])dµ (i). Recall µ is the family of conditional measures − p,α(i) 1 p { p,α(i)} of µ deﬁned by the measurable partition α(i) = Π 1(Π(i)). p R − Moreover, if the IFS satisﬁes the ROSC and p , then H = 0. ∈ P0

2.4 Upper bound for dimH Λ

Consider a shifted triangular Gatzouras–Lalley-type planar carpet Λ without any separation condition. To prove Theorem 2.2.1 we essentially lift the original argument in [GL92], formulated on the attractor Λ, to the symbolic space Σ. This can be done because the method in [GL92] is completely symbolic in nature. Therefore, we only give a short sketch. The first step is to define a proper metric on Σ, which captures the distance between points on the attractor. Observe that for two points i, j Σ the distance ∈ Π(i) Π(j) (recall (2.3.2)) can be small even if i j is small. This occurs if Φ(i) | − | | ∧ | | ∧ Φ(j) = ıˆ ˆ (recall (2.3.1)) is much larger than i j , i.e. the corresponding cylin- | | ∧ | | ∧ | ders belong to the same column for a long time. Lemma 2.4.1. (Σ, d) is a metric space, where the distance between i, j Σ is defined ∈ ıˆ ˆ i j | ∧ | | ∧ | d( ) = b + a i, j : ∏ ik ∏ ik . k=1 k=1 36 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

Proof. The fact that d is non-negative and symmetric is trivial. Need to check the triangle inequality, for all i, j, k Σ : d(i, j) d(i, k) + d(k, j). If ∈ ≤ i k i j i k i j then ∏| ∧ | a ∏| ∧ | a , • | ∧ | ≤ | ∧ | `=1 i` ≥ `=1 i` k j i j i k > i j then k j = i j , thus ∏| ∧ | a = ∏| ∧ | a . • | ∧ | | ∧ | | ∧ | | ∧ | `=1 i` `=1 i` Analogously, if

ıˆ kˆ ıˆ ˆ ıˆ kˆ ıˆ ˆ then ∏| ∧ | b ∏| ∧ | b , • | ∧ | ≤ | ∧ | `=1 i` ≥ `=1 i` kˆ ˆ ıˆ ˆ ıˆ kˆ > ıˆ ˆ then kˆ ˆ = ıˆ ˆ , thus ∏| ∧ | b = ∏| ∧ | b . • | ∧ | | ∧ | | ∧ | | ∧ | `=1 i` `=1 i` The triangle inequality now follows.

Remark 2.4.2. Without difﬁculty, one can show the stronger assertion that d(i, j) is an ul- trametric on Σ, i.e. d(i, j) max d(i, k) , d(k, j) for every i, j, k Σ. ≤ { } ∈ The next step is to prove that the natural projection with this metric is Lipschitz.

Lemma 2.4.3. For any shifted triangular Gatzouras–Lalley-type planar carpet

dim Λ dim (Σ, d). H ≤ H Proof. It is enough to show that there exists C > 0 such that Π(i) Π(j) C | − | ≤ · d(i, j), i.e. Π : Σ Λ is a Lipschitz-function, which can not increase the Hausdorff → dimension. For i, j Σ let k := i j , ` := ıˆ ˆ and ∈ | ∧ | | ∧ | ∞ x ai n 1tin,1 aj n 1tjn,1 := Π(i) Π(j) = ∑ | − − | − . y − di n 1tin,1 dj n 1tjn,1 + bi n 1tin,2 bj n 1tjn,2 n=1 | − − | − | − − | −

The ﬁrst k terms coincide in both coordinates and bin = bjn for n = 1, . . . , `. Thus,

∞ 2 2 2 2 x = ai k ∑ (aσki n k 1tin+k,1 aσkj n k 1tjn+k,1) ai k , | · | − − − | − − ≤ | n=k+1 ∞ 2 ∞ 2 ∞ 2 2 y 2 di n 1tin,1 + dj n 1tjn,1 + (bi n 1tin,2 bj n 1tjn,2) . ≤ ∑ | − ∑ | − ∑ | − − | − h n=1 n=1 n=1 i

In the ﬁrst two sums using Lemma 2.1.5 we can bound di n 1 K0 bi n 1 and dj n 1 | − ≤ · | − | − ≤ K0 bj n 1. Now we can pull out bi ` from all three sums. The remaining sums are all · | − | uniformly bounded in i, j by some constant c. This gives

2 2 2 2 2 2 y 2K0 c bi `, thus Π(i) Π(j) ai k + 2K0 c bi ` C d(i, j). ≤ · · | | − | ≤ | · · | ≤ · q

It remains to show that the value α∗ maximizing the expression for D(p) in (2.2.1) is an upper bound for the Hausdorff dimension of (Σ, d).

Proposition 2.4.4. For any choice of parameters deﬁning a shifted triangular Gatzouras– Lalley-type triangular carpet dim (Σ, d) α∗. H ≤ 2.4. Upper bound for dimH Λ 37

Proof of Theorem 2.2.1. The upper bound is a corollary of Lemma 2.4.3 and Proposition 2.4.4. The compactness of and the continuity of D(p) implies that sup D(p) is P p attained for some p . Moreover, it is easy to check that p , see [GL92, ∗ ∈ P ∗ ∈ P0 Proposition 3.4].

Proposition 2.4.4 is essentially proved in [GL92, Section 5] formulated on the attractor Λ. For completeness we sketch the main steps adapted to (Σ, d) and cite [GL92] when necessary. Most of the notation we bring over from [GL92]. The balls in (Σ, d) are exactly the "approximate squares" deﬁned in [GL92, eq. (1.2)]

B (i) := j Σ : i j L (i) and ıˆ ˆ k , where (2.4.1) k { ∈ | ∧ | ≥ k | ∧ | ≥ } k n L (i) := max n 0 : b a . k ≥ ∏ ij ≤ ∏ ij n j=1 j=1 o Note, k > L (i) for every i and k, since a < b for every i. The j B (i) for which k i i ∈ k i j = L (i) and ıˆ ˆ = k are the ones for which d(i, j) is maximal. The deﬁnition | ∧ | k | ∧ | of Lk(i) implies that

Lk(i) a ∏j=1 ij 1 1 max ai− for every k, i. (2.4.2) ≤ k ≤ i ∏j=1 bij

Hence, diamB (i) C ∏k b for some C independent of i. k ≤ · j=1 ij The main ingredient is a form of the mass distribution principle adapted to (Σ, d).

Lemma 2.4.5. Let µ be a probability measure on Σ and assume

log µ(B (i)) lim inf k α for every i Σ, k ∞ k ≤ ∈ → log ∏j=1 bij where Bk(i) is the approximate square deﬁned in (2.4.1). Then

dim (Σ, d) α. H ≤ Proof. The assumption states that for every ε, δ > 0 and i Σ there exists a k(i) such ∈ that k(i) k(i) α+ε b < δ and b µ(B (i)). ∏ ij ∏ ij ≤ k(i) j=1 j=1 The collection Bk(i)(i) i Σ is a δ-cover of Σ, thus the Vitali- or 5r-covering lemma { } ∈ [Fal86] implies that there exists a (perhaps uncountable) sub-collection J Σ of dis- ⊂ joint balls Bk(i)(i) giving a 5δ-cover of Σ, i.e.

Σ 5Bk(i)(i) and Bk(i)(i) Bk(j)(j) = ∅ for every i = j J. ⊆ i J ∩ 6 ∈ G∈ Hence, we can bound the α + ε-dimensional Hausdorff measure

k(i) α+ε α+ε α+ε α+ε α+ε 5δ (Σ) (5c) ∑ ∏ bij (5c) ∑ µ(Bk(i)(i)) (5c) µ(Σ) H ≤ i J j=1 ≤ i J ≤ · ∈ ∈ 38 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps independent of δ and therefore α+ε(Σ) (5c)α+ε < ∞ for every ε > 0. Thus, H ≤ dim (Σ, d) α. H ≤ The lemma implies that to prove Proposition 2.4.4 it is enough to ﬁnd a measure µ satisfying the condition of the lemma with the value α∗. This can be achieved using the family of Gatzouras–Lalley Bernoulli measures introduced in [GL92, eq. (5.2)]. Let ϑ R, λ R and ρ (0, 1). Deﬁne the probability vector p = (p ,..., p ) by ∈ ∈ ∈ 1 N ϑ λ ϑ ρ 1 ϑ pi = pi(ϑ, λ, ρ) := C(ϑ, λ, ρ)ai bi − (γi(ϑ)) − , where γi(ϑ) = ∑ aj (2.4.3) j ∈Iφ(i) and C(ϑ, λ, ρ) normalizes so that ∑i pi = 1. In fact [GL92, Lemma 5.1] shows that there exists a real-valued continuous function ϑ(ρ), ρ (0, 1), such that for every ∈ ρ (0, 1) ∈ C(ϑ(ρ), α∗, ρ) = 1. From now we work with such p.

Lemma 2.4.6. The Bernoulli-measure µ := pN on Σ satisﬁes the condition of Lemma 2.4.5 with the optimal value α∗, i.e.

µ(Bk(i)) lim inf α∗ for every i Σ. k ∞ k ≤ ∈ → log ∏j=1 bij

Sketch of proof. By deﬁnition of Bk(i)

Lk(i) ϑ k ρ Lk(i) k k α a ∗ ∏j=1 ij ∏j=1 γij (ϑ) µ(Bk(i)) = pij qφ(i ) = bij , ∏ ∏ j ∏ k ϑ Lk(i) j=1 · j=L (i)+1 j=1 · ∏j=1 b · ∏ γ (ϑ) k ij j=1 ij

k where qφ(i ) = ∑` p`. Taking logarithm and dividing by log ∏j=1 bij gives j ∈Iφ(ij)

ρ k γ (ϑ) ∏j=1 ij Lk(i) k log L (i) ϑ log( a / b ) ∏ k γ (ϑ) log µ(Bk(i)) ∏j=1 ij ∏j=1 ij j=1 ij k = α∗ + k + k . log ∏j=1 bij log ∏j=1 bij log ∏j=1 bij

Due to (2.4.2), the second term tends to zero as k ∞. We can increase the third term → by replacing the denominator with k log min b . Hence, it is enough to prove that · i i there exists ρ (0, 1) such that for ϑ = ϑ(ρ) ∈ ρ k 1 Lk(i) lim sup ∑ log γij (ϑ) ∑ log γij (ϑ) 0. k ∞ k j=1 − k j=1 ≥ → This is exactly the statement in [GL92, eq. (5.10)]. For details see [GL92, pg. 565- 566].

Proof of Proposition 2.4.4. The Proposition is a direct corollary of Lemmas 2.4.5 and 2.4.6. 2.5. Proof of Theorem 2.2.2 39

2.5 Proof of Theorem 2.2.2

Our goal is to show that the Ledrappier–Young formula (2.3.8) of [BK17] for dimH νp, cited in Theorem 2.3.4, always equals the formula for D(p) in (2.2.1) under the conditions of Theorem 2.2.2. For the rest of this proof, we fix a p and assume ∈ P0 H satisfies Hochman’s Exponential Separation Condition and either each column independently satisfies the ROSC or Λ satisfies transversality and (2.2.2). The entropy of the system is h = log p (recall (2.3.6)), the Lyapunov-exponents µp − h ip from Lemma 2.3.3 are χ2 = log a and χ1 = log b . Hochman’s Exponential νp − h ip νp − h ip Separation Condition for implies No Dimension Drop for ν , recall (2.1.20), hence H q dim ν = log q / log b . As a result, to prove the theorem it is enough to show H q h iq h ip that the integral

H = log µα(i)([i1])dµp(i) = 0, − Z where µ is the family of conditional measures of µ deﬁned by the measurable { α(i)} p partition α(i) = Π 1(Π(i)) , recall (2.3.5). Since log µ ([i ]) 0, we have that { − } − α(i) 1 ≥ H = 0 if and only if µα(i)([i1]) = 1 for µp-a.a. i. (2.5.1)

Thus, it sufﬁces to show that µα(i) is concentrated on i for µp-typical i. Overlaps arising from the translations of columns or from intersections within a column can in theory cause problems. However, the next two results ensure that there is a full measure subset of Σ for which µα(i) is a point mass distribution. 1 Recall from (2.3.4) that β(i) = Φ− Φ(i) is the ’symbolic column’ if i. The ﬁrst claim ensures that there is a full measure subset Σ Σ where the translations of 1 ⊂ complete columns have no effect.

Claim 2.5.1. Assume Weak Almost Unique Coding holds for Σ , recall Deﬁnition 2.1.9. Then there exists a full measure subset Σ Σ such that for all i ΣH and for all (ˆ ,..., ˆ ) = 1 ⊂ ∈ 1 1 n 6 (ıˆ1,..., ıˆn) µα(i)([j1,..., jn]) = 0, (2.5.2) where φ(ik) = ıˆk and φ(jk) = ˆk for k = 1, . . . , n, recall (2.1.9) for the deﬁnition of φ. Consequently, for every i Σ we have ∈ 1 c µα(i)(β(i) ) = 0. (2.5.3)

The second claim deﬁnes the full-measure set Σ Σ where intersections within 2 ⊂ columns have no effect.

Proposition 2.5.2. Assume that the conditions of Theorem 2.2.2 hold. Then there exists a Σ Σ, with µ (Σ ) = 1 such that for every i Σ and k i 2 ⊂ p 2 ∈ 2 ∈ Iφ(i1) \ { 1} µ (β(i) α(i) [k]) = 0. α(i) ∩ ∩ Theorem 2.2.2 is a corollary of these two results. Sometimes we use the following notation:

Deﬁnition 2.5.3. Let F Σ be a subset of full measure. Then we deﬁne ⊂ F := i F : µ (Σ F) = 0 . ∈ α(i) \ n o Since µp(F) = 1, the disintegratione formula (2.3.5) implies that µp(F) = 1.

e 40 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

2.5.1 The proof of Theorem 2.2.2 assuming Claim 2.5.1 and Proposition 2.5.2 Proof of Theorem 2.2.2 assuming Claim 2.5.1 and Proposition 2.5.2. As we established above in (2.5.1) that to prove the theorem it is enough to check that

µ (α(i) [i ]c) = 0, for µ-a.a. i. (2.5.4) α(i) ∩ 1 Clearly,

α(i) [i ]c (α(i) [k]) (α(i) [k]) . ∩ 1 ⊂ ∩ ∪ ∩ k ! k i ! 6∈I[φ(i1) ∈Iφ([i1)\{ 1} It follows from (2.5.2) that for every i Σ ∈ 1

µα(i) (α(i) [k]) = 0, (2.5.5) k ∩ ! 6∈I[φ(i1) where Σ1 is deﬁned in Claim 2.5.1. Thus, to prove the theorem we only need to verify that µ (α(i) [k]) = 0 for µ-a.a. i. (2.5.6) α(i) ∩ k i ! ∈Iφ([i1)\{ 1} We can write

(α(i) [k]) Σc Σc ∩ ⊂ 1 ∪ 2 k i ∈Iφ([i1)\{ 1}

(α(i) β(i) [k]) (α(i) β(i)c [k]) . ∪ ∩ ∩ ∪ ∩ ∩ k i ! k i ! ∈Iφ([i1)\{ 1} ∈Iφ([i1)\{ 1} U V

It follows from| Proposition{z 2.5.2 that µ } (U)| = 0 for all i {z Σ and it follows} α(i) ∈ 2 from Claim 2.5.1 that µ (V) = 0 for all i Σ . So, for all i Σ Σ (2.5.6) α(i) ∈ 1 ∈ 1 ∩ 2 holds, which together with (2.5.5) yields that (2.5.4) holds. This completes the proof of Theorem 2.2.2 assuming Claim 2.5.1 and Proposition 2.5.2.

2.5.2 The proof of Claim 2.5.1 Proof of Claim 2.5.1. In the definition of Weak Almost Unique Coding, recall Defini- tion 2.1.9, there is a set Σ defined in such a way that for Σ0 := Σ we BH ⊂ H H H \BH have µq(Σ0 ) = 1 and H 1 ıˆ Σ0 Σ0 Π− Π (ıˆ) = ıˆ , ∈ H ⇐⇒ H ∩ H H { } where Π is the natural projection from Σ to Λ . Let H H H 1 1 := Φ− ( ) and Σ0 := Φ− Σ0 . B BH H 2.5. Proof of Theorem 2.2.2 41

Since µq(Σ0 ) = 1 we can define Σ1 := Σ0 (recall the notation from Definition 2.5.3) H so that µp(Σ1) = 1 and µ ( ) = 0e for all i Σ . (2.5.7) α(i) B ∈ 1 e Recall Π is the natural projection from Σ to Λ . Observe that by definition H H 1 e i Σ0 = Σ0 Π− Π (i) = β(i). (2.5.8) ∈ ⇒ ∩ H H 1 Since α(i) Π− Π (i), we get from (2.5.8e) thate i Σ0 = Σ0 α(i) β(i). Equiva- ⊂ H ∈ ⇒ ∩ ⊂ lently, H e e i Σ0 = α(i) β(i) . ∈ ⇒ ⊂ ∪ B By definition

[j ,..., j ] β(i) = ∅ iff (ˆ ,..., ˆ ) = (ıˆ ,..., ıˆ ). 1 n ∩ 1 n 6 1 n That is for i Σ whenever (ˆ ,..., ˆ ) = (ıˆ ,..., ıˆ ) then [j ,..., j ] α(i) . So, ∈ 1 1 n 6 1 n 1 n ∩ ⊂ B (2.5.7) implies that (2.5.2) holds. To obtain (2.5.3) from (2.5.2), we write β(i)c as a countable union

∞ ∞ β(i)c = j Σ : ıˆ ˆ = ` = [j ,... j ] . { ∈ | ∧ | } 1 `+1 `=0 `=0 ˆr=ıˆr,r ` [ [ j [=i ≤ `+16 `+1 By (2.5.2) the measure of each cylinder of the right hand side is

µ ([j ,... j ]) = 0 if [ˆ ,... ˆ ] = [ıˆ ,..., ıˆ ] , i Σ . α(i) 1 `+1 1 `+1 6 1 `+1 ∈ 1

2.5.3 Proof of Proposition 2.5.2 If the columns independently satisfy ROSC, then the proof of [BK17, Corollary 2.8] can be repeated in this setting, therefore we omit it. In the remainder we assume the shifted TGL carpet Λ satisﬁes transversality and (2.2.2):

log a p log N q h i > 1 + h i . log b log q h ip − h iq Throughout this proof we ﬁx δ > 0 small enough such that

(1 + δ) log N q log a p 1 + δ + h i < (1 δ) h i . (2.5.9) δ log b log q − log b h ip − h iq h ip This can be achieved since the expression is continuous in δ and we assume (2.2.2). The reason that we require this is that for such a δ and

log a p u := (1 δ) h i (1 + δ), (2.5.10) − log b − h ip the inequality in (2.5.9) is equivalent to

(1+δ) u δu N q b − < 1. (2.5.11) h i · h i · h i 42 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

At the very end of this proof we will need this. The importance of the u defined above comes from the fact that for an arbitrary ` and k = u `, · (1 δ)` k a p − b p = h i . (2.5.12) h i b (1+δ)` h ip 1 1 Recall α(i) = Π− Π(i), β(i) = Φ− Φ(i), that Π is the natural projection from Σ to H Λ and that in (2.3.7) we define Badδ,n(c) for a c = (c1,..., cN) with c p = 1. H e h i 6 Further notation Recall that Hochman’s Exponential Separation Condition implies that for the self- similar measure νq on Λ we have dimH νq = log q q/ log b p. Feng and Hu [FH09] H h i h i proved that νq is exact dimensional. That is for K1 defined in (2.1.14) and

S := i Σ : n n , n0 ∈ ∀ ≥ 0 n n n n δn n δn νq Π (i) 3K1 b , Π (i) + 3K1 b q b , q b − (2.5.13) H − h i H h i ∈ h i h i h i h i o we havee e ∞

µp Sn0 = lim µp (Sn0 ) = 1. (2.5.14) ! n0 ∞ n[0=1 → We deﬁne the set of symbols which are "good" from level m on:

c Goodm := Badδ,n(a) Badδ,n(b) Badδ,n(N) Badδ,n(q) . n m ∪ ∪ ∪ \≥ Note that it follows from Lemma 2.3.1 that for

∞ Good := Goodm, we have µp(Good) = 1. (2.5.15) m[=1 To measure vertical distance and neighborhood on Λ we deﬁne

y y0 , if x = x0; disty((x0, y0), (x, y)) := | − | (∞. otherwise,

For every m 1 the function L : Good [0, 1] is deﬁned as follows: if there exists ≥ m → no j Good with j = i and Φ(i) = Φ(j) then L (i) := 1. Otherwise we deﬁne ∈ m 1 6 1 m L (i) := inf dist (Π(i), Π(j)) : j Good β(i) such that j = i . m { y ∈ m ∩ 1 6 1} Let

Vm := i Good : L (i) < a ` ` { ∈ m h i } = i Good : j β(i) [i ]c Good , dist (Π(i), Π(j)) < a ` . ∈ ∃ ∈ ∩ 1 ∩ m y h i n o Also deﬁne

m := i Good : L (i) = 0 = i Good : α(i) β(i) [i ]c Good = ∅ . B2 { ∈ m } { ∈ ∩ ∩ 1 ∩ m 6 } (2.5.16) 2.5. Proof of Theorem 2.2.2 43

m m+1 m m Clearly, 2 2 since 2 = ` mV` . The key lemma states the following. B ⊂ B B ∩ ≥ Lemma 2.5.4. For arbitrary m 1 we have µ( m) = 0. ≥ B2 Proof of Proposition 2.5.2 assuming Lemma 2.5.4. Let

∞ := m = i Good : α(i) β(i) [i ]c Good = ∅ . B2 B2 { ∈ ∩ ∩ 1 ∩ 6 } m[=1

c By Lemma 2.5.4, µ( 2) = 0. That is, if i Σ2 := Good^ 2 then on the one hand c B ∈ ∩ B µ (Good ) = 0, on the other hand α(i) β(i) [i ]c Good = ∅. This implies that α(i) ∩ ∩ 1 ∩ µ (α(i) β(i) [i ]c) = 0, which completes the proof of Proposition 2.5.2. α(i) ∩ ∩ 1 It remains to show Lemma 2.5.4. The method of the proof was inspired by [BRS, Lemma 4.7], however there are signiﬁcant differences. On the one hand, in [BRS] the q measure corresponding to νq is absolutely continuous with L density and in [BRS] the diagonal part of all the linear parts of all the mappings are identical. These differences required a much more subtle argument in this situation.

Proof of Lemma 2.5.4. Recall that we ﬁxed an m. Let ` m. All sets and numbers from ≥ now on in this proof can be dependent of m but m is ﬁxed so we omit it from notation. m 1 We cover V` by the union of the Π− pre-images of the parallelograms like the blue one (Rı,) on the right hand side of Figure 2.3. These are parallelograms slightly bigger than the intersection of R and the a ` neighborhood of R for ı,  Σ with ı h i  ∈ ` Φ(ı) = Φ(). To control the size of `th level parallelograms and the number of parallelograms in any given `-th level column set

1 1, 1 Bad := Bad (a) Bad (b) Bad (N) and Bad ∗ := i ` : i Bad , δ,` δ,` ∪ δ,` ∪ δ,` δ,` { | ∈ δ,`} 1 where Badδ,n(c) was deﬁned in (2.3.7). Observe that Badδ,` is the union of complete `-cylinders. That is 1 Badδ,` = [ω]. 1, ω Bad ∗ ∈[ δ,` The level `-cylinders of the symbolic spaces excluding these bad cylinders are:

` 1, 1, Good∗ := 1, . . . , N Bad ∗ and Good∗ :=  Good∗ : j = i , Φ() = Φ(ı) Bad ∗. ` { } \ δ,` `,ı { ∈ ` 1 6 1 }\ δ,` For H [0, 1]2 let ⊂ U (H, r) := (x, y) : x = x and y y < r . y { 0 | − 0| } (x ,y ) H 0 [0 ∈

Choose ı Good∗,  Good∗ and deﬁne ∈ ` ∈ `,ı (1 δ)` I := proj (R U (R , a − ), ı, x ı ∩ y  h i R := (I [0, 1]) R , ı, ı, × ∩ ı 1 R := ([ı] Π− (R )). ı, ∩ ı, e 44 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

Rı, consists of those elements of Rı which are physically "too close" to R, see Figure m 2.3. As a result we get a cover of V` :

Vm Bad1 R . (2.5.17) ` ⊂ δ,` ∪ ı, ı Good∗  Good∗ ∈ [ ` ∈ [ `,ı e m 1 Namely, if i V` then either i Badδ,` or ı := i ` Good`∗. In the second case, there ∈ c ∈ | ∈ ` is a j β(i) [i ] Good with dist (Π(i), Π(j)) < a . Hence,  := j ` Good∗ . ∈ ∩ 1 ∩ m y h i | ∈ `,ı As a result, with these notations, we have i R . ∈ ı, e

(1, 1) R Rı

aı )  ı, H

Π( (1 δ)` a − h i f 1 a ı− (Rı,) (1 δ)` a − Iı, h i (0, 0) Jı, Rı,

FIGURE 2.3: Intersecting parallelograms Rı and R in the proof of Lemma 2.5.4.

If I = ∅, then there exists a non-empty interval J such that ı, 6 ı, 1 f − (R ) = J [0, 1]. ı ı, ı, × 1 1 With symbolic notation Hı, := Π− ( fı− (Rı,)) we can represent Rı, as the concatenation Rı, = ıHı,.e (2.5.18) 1 On the other hand, Hı, Π− Jı, . Hence, ⊂ H e 1 µp eHı, µp Π− (Jı,) = νq Jı, . (2.5.19) ≤ H To continue we give an upper bound foreνq Jı, .

Claim 2.5.5. Let ı Good`∗ and  Good`∗,ı and let k := u `, where u was defined in 1 ∈ ∈ · (2.5.10). If Π− (Jı,) Sk = ∅ (recall (2.5.13) for the definition of Sk), then H ∩ 6 k kδ e ν J q b − . q ı, ≤ h i · h i Proof of Claim 2.5.5. If I = ∅ then transversality (recall Definition 2.1.6) implies that ı, 6 (1 δ)` I < 3K a − . | ı,| 1 · h i 2.5. Proof of Theorem 2.2.2 45

1, This is the very important point where use that neither ı nor  are contained in Badδ,`∗. 1 (1 δ)` (1+δ)` Furthermore, f − expands along the x axis by a factor between b and b , ı h i− − h i− hence ` a 1 δ J < 3K h i − . (2.5.20) | ı,| 1 b 1+δ h i If we set k as in Claim 2.5.5 then as we mentioned in (2.5.12) the right hand side of (2.5.20) is less than 3K b k : 1 · h i J < 3K b k. | ı,| 1 · h i 1 1 Now assume that Π− (Jı,) Sk = ∅. Pick an arbitrary ω Π− (Jı,) Sk. Then H ∩ 6 ∈ H ∩ k k eJı, Π (ω) 3K1 b , Π (ω) + 3K1 be . ⊂ H − h i H h i k k k Using that ω Sk, we gete that νq Π (ω) e 3K1 b , Π (ω) + 3K1 b q ∈ H − h i H h i ≤ h i · b kδ. − h i e e

m Now we conclude the proof of Lemma 2.5.4. From the cover (2.5.17) of V` together with (2.5.18) we obtain that for ` m ≥

µ (Vm) µ (Bad1 ) + µ ([ı])µ H . (2.5.21) p ` ≤ p δ,` ∑ p p ı, ı Good`∗  Good∗ ! ∈ ∈ [ `,ı To further bound (2.5.21), ﬁrst observe that

(1+δ)` #Good∗ N whenever ı Good∗. `,ı ≤ h i ∈ `

Moreover, using (2.5.19) and Claim 2.5.5, for an arbitrary ı Good∗ we have ∈ `

c c µp Hı, µp (Sk) + ∑ µp Hı, µp (Sk) + ∑ νq(Jı,) ≤  Good ≤  Good  Good∗ ! `∗,ı `∗,ı ∈ [ `,ı ∈ ∈ Π 1(J ) S =∅ Π 1(J ) S =∅ − ı, ∩ k6 − ı, ∩ k6 H H c e k kδ e µp (Sk) + ∑ q b − ≤  Good h i · h i ∈ `∗,ı Π 1(J ) S =∅ − ı, ∩ k6 H e ` c (1+δ) u uδ µ (S ) + N q b − . ≤ p k h i · h i · h i Pluggung this back into (2.5.21), we deduce from Lemma 2.3.1,(2.5.14) and (2.5.11) that for every m, m lim µp(V` ) = 0. ` ∞ → By (2.5.16), this implies that µ ( m) = 0. p B2 46 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

2.6 Proof of results for box dimension

We begin the section by briefly commenting on how the upper bound for dimBΛ, recall Theorem 2.2.4, follows directly from the work of Fraser [Fra12] and then prove Theorem 2.2.7 in Subsection 2.6.3. Recall the notation from Subsection 2.2.2. For δ > 0 and a bounded set F R2 ⊂ let Nδ(F) denote the minimal number of closed axes parallel rectangles for which the vertical sides are not shorter than the horizontal sides but the vertical sides are not longer than (K0 + 1)-times the horizontal sides, where K0 was defined in Lemma 2.1.5. Then log Nδ(F) log Nδ(F) dimBF = lim inf and dimBF = lim sup . δ 0 log δ δ 0 log δ → − → − In particular, it is enough to consider δ 0 through the sequence δ = ck for some → k 0 < c < 1, see [Fal90, Section 3.1]. For t 0 and any finite length word ı Σ , Fraser defined the modified singular ≥ ∈ ∗ value function ψt, which in our context is

t s t s ψ ( f ) := b H a − H , ı ı · ı where s = dimB Λ . He showed that the unique solution s of the equation H H 1/n s lim ψ ( fı) = 1 n ∞ ∑ → ı Σn ∈ is an upper bound for dimBΛ and equals dimB Λ if Λ satisﬁes the ROSC. In our con- N s s s text this equation simply becomes (2.2.4): ∑i=1 bi H ai − H = 1. The slight modiﬁcation of the GL brother Λ ensures that the solution of (2.2.4) for Λ and Λ is the same. For any TGL carpet Λ, it follows from Lemma 2.1.5 that the longer side of any parallelogram R ise at most (K + 1) b . This implies that theree exists a constant C ı 0 · ı (independent of δ) such that N (Λ) C N (Λ). Hence, dim (Λ) dim (Λ) s. δ ≤ · δ B ≤ B ≤ Furthermore, when the ROSC is assumed, it is clear that the reversed inequalities also hold. This implies dim (Λ) dim (Λ) = s. Thise proves Theorem 2.2.4. e B ≥ B In the presence of overlaps, one must be more careful when counting the intersections. The next subsection shows howe to select a diagonally homogeneous subsystem from a higher iterate of . F 2.6.1 Diagonally homogeneous subsystems Recall for a p ∈ P0 N hp = ∑ pi log pi = log p p, − i=1 − h i N N 1 2 χp = ∑ pi log bi = log b p and χp = ∑ pi log ai = log a p. − i=1 − h i − i=1 − h i

The following is a Ledrappier–Young like formula for the solution s of (2.2.4). It gen- eralizes the formula in Corollary 2.2.12 for the diagonally homogeneous case. A similar result for Bedford-McMullen like systems in arbitrary dimension was proved in [FH09, Theorem 2.15]. 2.6. Proof of results for box dimension 47

s s s Claim 2.6.1. For p := (p1,..., pN) deﬁned by pi = bi H ai − H , recall (2.2.6), we have

1 e e e hp e χp s = + 1 s , χ2 − χ2 H ep pe ! where s = dimB Λ . e e H H 1 2 Proof. Immediately follows from the observation that hp = s χ + (s s )χ . H p − H p

The following line of thought is an adaptation ofe [FS16, Sectione 6] in ordere to extract from an arbitrary shifted TGL IFS = f N with M columns a subsystem F { i}i=1 of a high enough iterate of k, which has some nice properties required to prove the F theorem. Let k := f : i ,..., i Σ . The ﬁrst step is to pass from to a diagonally F { i1...ik 1 k ∈ k} F homogeneous subsystem of k. Analogous arguments appear for example in [BRS16, F Lemma 5.2], [FS16, Lemma 6.2] or [PS, Lemma 4.9]. Deﬁnition 2.6.2. A subsystem (k) k is called a diagonally homogeneous subsystem if G ⊂ F there exists a(k) and b(k) for which

(k) (k) b 0 (k) g (x) = x + t , for every g (k). i d a(k) i i ∈ G i

Fix an arbitrary vector v = (v1,..., vN), where vi N. Let Vˆ := ∑i vi, V := ∈ ∈Iˆ (V1,..., VM), V = V1 + ... + VM and deﬁne

:= (i ,..., i ) Σ :# ` V : i = r = v for every r = 1, . . . , N . (2.6.1) Mv { 1 V ∈ V { ≤ ` } r } Claim 2.6.3. The subsystem = f : (i ,..., i ) V with M columns Gv { i1...iV 1 V ∈ Mv} ⊂ F v (v) N vr (v) N vr (i) is a diagonally homogeneous subsystem with a = ∏r=1 ar and b = ∏r=1 br , (ii) has uniform vertical ﬁbres with ∏M Vıˆ! maps in each column and ıˆ=1 ∏r vr! ∈Iıˆ (iii) for the probability vectors v = v/V and V = V/V

N log(V + 1) + V h log # V h , − · v ≤ Mv ≤ · v N log(V + 1) + V h log M V h . − · V ≤ v ≤ · V Proof. Parts (i) and (ii) are immediate. Part (iii) follows directly from [DZ10, Lemma 2.1.8].

Lemma 2.6.4. Let = f N be a shifted TGL IFS with M columns. For every k choose F { i}i=1 vk = (v1,k,..., vN,k) such that

v = kp for every i = 1, . . . , N, (2.6.2) i,k b ic (k) (k) M (k) where pi was defined in (2.2.6). Lete Vˆ := ∑i ˆ vi,k, V = ∑ıˆ=1 Vıˆ and define the subsystem ∈I (k) V(k) e = vk = fi1...i (k) : (i1,..., iV(k) ) vk , G G { V ∈ M } ⊂ F where is defined by (2.6.1). Then (k) satisfies the assertions of Claim 2.6.3 with v . For Mvk G k brevity we write a(vk) = a(k) and b(vk) = b(k). Let

N(k) = # and M(k) = # Φ( ) Mvk Mvk 48 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps denote the number of maps and columns in (k). (k) (k) G (k) (k) s (k) s(k) s Moreover, lim s = s, where s is the solution of N (b ) H (a ) − H = 1, i.e. k ∞ → log N(k) log b(k) s(k) = + 1 s . log a(k) − log a(k) H − ! Remark 2.6.5. The box dimension of the attractor of the IFS (k) is NOT equal to s(k), because G s is not the box dimension s(k) of the attractor of the IFS generated by the columns of (k). H (k) H G The problem is that s s as k ∞ (except when dimH Λ = dimB Λ). H 6→ H → Proof. It follows from (2.6.2) that k N V(k) k and − ≤ ≤ N (k) k log a p ∑ log ai log a k log a p. h i − i=1 ≤ ≤ h i e e (k) (k) Same holds for log b . Furthermore, for the probability vector vk = vk/V

1 vi,k pi N pi pi + . − k ≤ V(k) ≤ k N − e Thus limk ∞ hv = hp. We cane use Claim 2.6.3e (iii) to bound log # v . Hence, putting → k M k together all the above we get e hp log b p lim s(k) = + 1 h i s , k ∞ log a p − log a p H → − eh i h i e which is equal to s due to Claim 2.6.1. e e

If (k) already has non-overlapping columns, then the rest of the construction is G ` not necessary. Otherwise, we can pass further to a subsystem (k,`) (k) by ` G ⊂ G throwing away "not too many" columns of (k) in order to ensure that (k,`) has G G non-overlapping columns. (k) Projecting (k) to the x-axis gives a subsystem of V G H (k) := hıˆ1...ıˆ (k) : there exists (i1,..., iV(k) ) vk s. t. Φ(i1 ... iV(k) ) = ıˆ1 ... ıˆV(k) , GH { V ∈ M } ( ) which has a total of M(k) maps, each with contracting ratio b(k). Observe that k G also satisfies Hochman’s Exponential Separation Condition, because this conditionH is assumed for and this property passes on to any subsystem. Hence, the Hausdorff H ( ) and box dimension of k satisfies GH (k) ( ) log M s k = . (2.6.3) (k) H log b − It follows from the definition of box dimension that for every ε > 0 there exists a ` subset of the columns of (k) , which are non-overlapping and have cardinality G (k) (k,`) (k) ` (s ε) (2.6.3) (k) ` (k) `ε M C (b ) − H − = C M b . (2.6.4) ≥ ε ε · 2.6. Proof of results for box dimension 49

This is the subsystem (k,`) which we will use in the proof of Theorem 2.2.7 under G ` condition (i). When condition (ii) of Theorem 2.2.7 is assumed we use (k,`) = (k) G G since in this case non-overlapping columns are assumed for Λ. Next, we present our argument to count the number of intersections within a column when Λ has non- overlapping columns.

2.6.2 Counting intersections ` Let be an arbitrary TGL IFS and (k,`) = (k) be the subsystem defined in F G G the previous subsection. Then (k,`) is diagonally homogeneous with main diagonal G ((b(k))`, (a(k))`), has uniform vertical fibres with (N(k)/M(k))` maps in each column and the columns are non-overlapping. For every f (k,`), ı can be written ı ∈ G ı = ı ı ... ı , where ı for j = 1, . . . , `. 1 2 ` j ∈ Mvk Let Σ(k,`) := ı : f (k,`) and for the rest of the subsection fix such an ı Σ(k,`). { ı ∈ G } ∈ Let (k,`) Σ∼ :=  =  ...  Σ : Φ() = Φ(ı) and  = ı , ı { 1 ` ∈ 6 } i.e. Σı∼ collects those  which belong to the symbolic column of ı. Recall Λı = 2 fı(Λ), Rı = fı([0, 1] ). Let

( ) R := proj ( f (Λ)) [0, 1] R and δ k := (a(k))`. ı x ı × ∩ ı ` N (k) R ( R ) Our aim is to givee a uniform upper bound for ı  Σı∼  . Observe that for δ` ∩ ∪ ∈ every  Σı∼ ∈ e e 1 N (k) Rı R = N (k) projx(Λı Λ) = N (k) (k) ` hı− (projx(Λı Λ)) . (2.6.5) δ` ∩ δ` ∩ δ` /(b ) ∩ We statee ae result of Lalley [Lal88, Theorem 1], which gives the precise asymptotic of Nδ(Λ ). A set r1,..., rM of positive reals is τ-arithmetic, if τ > 0 is the greatest H { } number such that each ri is an integer multiple of τ, and non-arithmetic if no such τ exists. We use the notation f (δ) g(δ) to denote that limδ 0 f (δ)/g(δ) = 1. Let F be ∼ → a self-similar set on [0, 1] with contracting ratios r1,..., rM . Assume F satisﬁes the { } M t strong OSC and let dimH F = dimB F = t, where t is the solution of ∑ıˆ=1 rıˆ = 1. 1 1 Proposition 2.6.6. [Lal88, Theorem 1] If log r− , . . . , log r− is a non-arithmetic set, then { 1 M } for some K > 0 t N (F) Kδ− as δ 0. δ ∼ → 1 1 On the other hand, if log r1− , . . . , log r−M is τ-arithmetic, then for the subsequence δn = nτ { } e− there exists a constant K0 > 0 such that

t N (F) K0δ− as n ∞. δn ∼ n → Remark 2.6.7. The reason why we can not handle both types of overlaps simultaneously for the box dimension is that we are unaware of an analogous result in the case that SOSC is not assumed. This question could be of independent interest.

We use the proposition for the self-similar set Λ with contracting ratios (r1,..., rM). 1 1 H If log r− , . . . , log r− is τ-arithmetic, then we can choose ` = `(n) so that { 1 M } τ τn (k) τ τn min e− , 1 e− < δ max e− , 1 e− , { }· ` ≤ { }· 50 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

which implies that

`(n) τ Ne τn (Λ ) lim = and lim − H = c (k) n ∞ n n ∞ N (k) ( ) → log a → Λ − δ` H for some universal constant c. Thus the proposition implies that

(k) (k) `s N (k) Rı = N (k) (k) ` (Λ ) = (C + o(1))(b /a ) H , (2.6.6) δ` δ` /(b ) H 1 1 where the constant Ceonly depends on whether log r− , . . . , log r− is τ-arithmetic { 1 M } or not and the o(1) 0 as ` ∞. The next lemma ensures that a positive proportion → → of these boxes do not get covered by boxes coming from the cover of R for some  Σı∼. ∈ e Lemma 2.6.8. If satisﬁes transversality and F s (k) (k) H N s b 1 + K1H < , (2.6.7) M(k) a(k) ! then there exists K < 1 such that for ` large enough and every ı Σ(k,`) we have 3 ∈

N (k) Rı R K3N (k) Rı . δ` δ` ∩  Σ ≤ ∈[ı∼ e e e Proof. Fix  Σ(k,`) such that  ı = z, where we count ı ,  as one sym- ∈ | ∧ | m m ∈ Mvk bol. Thus, z 0, 1, . . . , ` 1 . Since satisﬁes transversality, then so do all of its ∈ { − } F subsystems, in particular (k,`) as well. Hence, G z ` z proj R R K b(k) a(k) − , | x ı ∩  | ≤ 1 see Figure 2.4. This together withe (2.6.5e ) and Proposition 2.6.6 yields that zs (k) H s b N (k) (R R ) (C + o(1))K H . ı  1 (k) δ` ∩ ≤ a ! e e (k,`) Since has uniform vertical ﬁbres, it follows that #  Σı∼ :  ı = z (k) G (k) ` z { ∈ | ∧ | } ≤ (N /M ) − . Thus from a simple union bound we get

` z zs ` 1 (k) − (k) H − N s b N (k) Rı R (C + o(1))K1H δ` ∑ (k) (k) ∩  Σ ≤ z=0 M ! a ! ∈[ı∼ e e `s ` s (k) H (k) K1H b M = (C + o(1)) K3N (k) Rı , M(k) b(k) s (k) (k) δ H  a − N  ≤ ` (k) (k) 1 ! ! N a −   e =:K3 | {z } where the last inequality holds if N(k)/M(k) (b(k)/a(k))s . This holds, because ≤ H (2.6.7) is an even stronger assumption. Furthermore, simple arithmetic shows that K3 < 1 if and only if (2.6.7) holds. 2.6. Proof of results for box dimension 51

(a(k))`

Rı (a(k))`

(a(k))` R

(k) ` z ` z (b ) proj R R K b(k) a(k) − | x ı ∩  | ≤ 1 e e FIGURE 2.4: Intersecting parallelograms Rı and R in the proof of Lemma 2.6.8.

2.6.3 Proof of Theorem 2.2.7 Throughout the proof, s is the target box dimension deﬁned as the solution of (2.2.4): N s s s (k,`) ∑ b H a − H = 1. Fix ε > 0. We work with the subsystem deﬁned in Subsec- i=1 i i G tion 2.6.1. It will be enough to cover the subset

f (Λ) Λ, ı ⊆ ı (k,`) ∈G[ ( ) with boxes of size δ k := (a(k))`. Recall R = (proj ( f (Λ)) [0, 1]) ( f ([0, 1]2)). ` ı x ı × ∩ ı Conclusion of proof assuming condition (i) of Theorem 2.2.7. Assume generates a shifted e F TGL carpet Λ for which satisﬁes Hochman’s Exponential Separation Condition and H the columns independently satisfy ROSC. In this case it is enough to use the deﬁni- (k) (k) `(s ε) tion of box dimension to bound N (k) Rı Cε(b /a ) H− for some constant Cε δ` ≥ depending only on ε. Recall from Lemma 2.6.4 that s(k) s. We choose k so large e → that s(k) s ε and we bound ≥ −

log N (k) (Λ) (k,`) (k) (k) ` (k) (k) `(s ε) δ log M (N /M ) (b /a ) H− lim inf ` lim inf (2.6.8) (k) (k) ` ∞ log δ ≥ ` ∞ ` log a → − ` → − log N(k) log b(k) + 1 s ε s 2ε, ≥ log a(k) − log a(k) H − ≥ − − ! =s(k) s ε ≥ − | {z } where for the second inequality we substituted the lower bound for M(k,`) from (2.6.4). Letting ε 0 yields dim Λ s as claimed. & B ≥ Conclusion of proof assuming condition (ii) of Theorem 2.2.7. For the remainder we assume that has non-overlapping columns, satisﬁes transversality and (2.2.5): F

hp hq < s (log b p log a p), (2.6.9) − H h i − h i e e s s s e e where h = log p and p = b H a − H . We need to check that condition (2.6.7) p − h ip i i i of Lemma 2.6.8 is satisﬁed, since it ensures that a positive proportion of the boxes needed toe cover f (Λ) eare not intersected by any boxes covering f (Λ) for  = ı. ı e e  6 52 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

Claim 2.6.9. For all k large enough, assumption (2.6.9) implies condition

s (k) (k) H N s b 1 + K1H < M(k) a(k) ! of Lemma 2.6.8.

Proof of Claim 2.6.9. We know from Subsection 2.6.1 that

log a(k) = k log a + O(1), log N(k) = kh + o(k), h ip p log b(k) = k log b + O(1), log M(k) = kh + o(k). h iep eq Taking the logarithm of each side ofe (2.6.7), substituting thesee values and dividing by k gives 1 s hp hq + log(1 + K H ) < s (log b p log a p), − k 1 H h i − h i with an error of o(e1) asek ∞ on either side. The seconde term one the left hand side → also tends to zero as k ∞, thus (2.6.9) indeed implies the condition of Lemma 2.6.8 → for large k.

The conclusion of the proof of Theorem 2.2.7 is now analogous to the calculation of (2.6.8) with the exception that we need the precise value of N (k) Rı from (2.6.6) δ` and we can use (k,`) = ( (k))`, so the number of columns M(k,`) = (M(k))`. Choose k G G e so large that s(k) s ε and condition (2.6.7) hold simultaneously. Using Lemma 2.6.8 ≥ − we can basically repeat the calculation of (2.6.8)

log N (k) (Λ) (k) ` (k) (k) `s δ log (N ) (1 K3)(C + o(1))(b /a ) H lim inf ` lim inf − = s(k). (k) (k) ` ∞ log δ ≥ ` ∞ ` log a → − ` → − This concludes the proof of Theorem 2.2.7.

2.6.4 Proof of Theorem 2.2.8 The theorem claims that for a shifted TGL carpet Λ

s s (i) dimH Λ = dimB Λ (ii) s = dimH νq (iii) ∑ aj− H = 1 for every ıˆ [M] H j ∈ ∈Iıˆ e are equivalent, provided ROSC and No Dimension Drop (NDD, recall Definition 2.1.9) hold. We show that (i) (iii), (iii) (ii) and (ii) (i). ⇔ ⇒ ⇒ Proof of (i) (iii). Let Λ be the GL brother of Λ, recall Definition 2.1.4. For ⇔ a p let ν denote the push forward of the Bernoulli measure µ on Λ. We ∈ P0 p p have dim ν = dim ν for everye p . Indeed, in the beginning of Section 2.5 H p H p ∈ P0 we proved dimeH νp = D(p) assuming ROSC and NDD, furthermore, Gatzouras–e Lalley proved dimH νp e= D(p) [GL92, Proposition 3.3]. Hence, dimH Λ = dimH Λ. M s Also, assuming NDD, s is the unique real which satisfies ∑ıˆ=1 rı H = 1. This implies H ˆ dim Λ = dim Λ. Thee analogous claim of (i) (iii) for Λ was proved in [GL92e, B B ⇔ Theorem 4.6]. Thus (i) (iii) in our setting as well. ⇔ s Proof of (iii) e (ii). Condition (iii) implies that the vectore q is simply qıˆ = rıˆH ⇒ 1 for ıˆ [M], where rıˆ = bj if j ıˆ. NDD is assumed, thus dimH νq = hq/χq = 1 ∈ 1 ∈ I s χq/χq = s . e e H H e e e

e e 2.7. Examples 53

Proof of (ii) (i). We can use Claim 2.6.1 and (2.3.8) to see that ⇒ 1 2 0 dimB Λ dimH Λ dimB Λ dimH νp = 1 χp/χp s dimH νq . ≤ − ≤ − − H − e e e e Clearly, (ii) implies dimH Λ = dimB Λ. This concludes the proof of Theorem 2.2.8.

2.7 Examples

We now treat the examples presented in Subsection 1.4.1 in detail. We do not calculate numerically the exact value of the dimensions for the TGL carpet of Figure 1.6, rather just comment why dimH Λ < dimB Λ < dimAff Λ. It satisﬁes the ROSC, thus its dimensions are equal to its GL brother. Clearly, the IFSs on [0, 1] generated from a vertical line in each of the columns do not have the same dimension. Hence, the third condition of (2.2.7) of Theorem 2.2.8 does not hold. Fur- thermore, dimB Λ < 1 because there is an empty column. Thus, Corollary 2.2.5 H implies that dimB Λ < dimAff Λ. Except for the "X X" example, all the other ones of Subsection 1.4.1 satisfy ≡ Λ = [0, 1], hence Corollary 2.2.5 implies dimB Λ = dimAff Λ. H 2.7.1 The self-afﬁne smiley: a non diagonally homogeneous example The smiley is constructed from the TGL IFS

8 b 0 = fi(x) = x + ti , F di ai i=1 where b = 0.2, a1 = ... = a5 = 0.1, a6 = a7 = a8 = 0.13 and the off-diagonal elements d = 0.2, d = 0.1, d = d = d = 0, d = 0.1, d = d = 0.2. The trans- 1 − 2 − 3 7 8 4 5 6 lations were chosen so that the mouth is constructed from f1,..., f5, the nose from f6 and the eyes from f7 and f8. It is non diagonally homogeneous since the mouth is thinner than the nose and eyes. Clearly, Λ does not have uniform vertical ﬁbres, thus Theorem 2.2.8 implies dimH Λ < dimB Λ. The numerical values of the dimensions given in Figure 1.7 were obtained using Wolfram Mathematica 11.2. The box dimen- N s s s sion was calculated from ∑i=1 bi H ai − H = 1, recall (2.2.4), while the maximization of D(p) (2.2.1) gave the Hausdorff dimension.

2.7.2 Example for dimH Λ = dimB Λ Deﬁne the matrices 1/3 0 1/3 0 1/3 0 A := , A := , A := . 1 0 a 2 1/2 a a 3 a 1/2 a − − For a (0, 1/3) deﬁne the IFS consisting of ∈ Fa 1/3 1/3 0 f (x) = A x + , f (x) = A x + , f (x) = A x + , 1 1 0 2 1 1 a 3 2 1/2 − 2/3 0 2/3 f (x) = A x + , f (x) = A x + , f (x) = A x + . 4 2 0 5 3 1/2 a 6 3 1 a − −

The attractor Λa is shown in Figure 1.8 for a = 3/10. Falconer and Miao showed in [FM07] how to calculate the box dimension and later Bárány in [B15´ ] showed that 54 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

the same value is a lower bound for the Hausdorff dimension. Hence, dimH Λa = dimB Λa. Alternatively, we can now argue that Λa is a diagonally homogeneous TGL carpet for every a (0, 1/3) satisfying ROSC with uniform vertical ﬁbres. Hence, our results ∈ apply. After some basic arithmetic, the dimension formula simpliﬁes to

log 2 dim Λ = dim Λ = 1 . (2.7.1) H a B a − log a

2.7.3 Overlapping example With a modiﬁcation of the translation vectors in the previous example, we construct a carpet with overlapping cylinders, see Figure 1.8. Deﬁne

1/3 1/3 0 f (x) = A x + , f (x) = A x + , f (x) = A x + , 1 1 1/4 2 1 3/4 a 3 2 1/4 − 2/3 0 2/3 f (x) = A x + , f (x) = A x + , f (x) = A x + , 4 2 1/4 5 3 3/4 a 6 3 3/4 a − − where the matrices A , A and A are from Subsection 2.7.2. For a (0, 1/3) the 1 2 3 ∈ attractor Λa is a diagonally homogeneous TGL carpet with uniform vertical fibres and non-overlapping columns. Transversality must be satisfied in order to apply our results. It would suffice to check (2.1.16) in Lemma 2.1.8, but in fact the constant K1 in Definition 2.1.6 of transversality can be directly bounded in this example.

Claim 2.7.1. Transversality holds for every a < 1/6 with

1/9 a/3 K < − . 1 (1/2 a)(1/3 2a) − − Proof. For brevity we write d := 1/2 a and b = 1/3. Let ı and  be two words − of length n such that i = j and φ(i ) ... φ(i ) = φ(j ) ... φ(j ). Since R R = ∅ 1 6 1 1 n 1 n ı ∩  6 and due to the symmetry in the construction, we may assume i1 = 3 and j1 = 5, hence d = d. A simple geometric exercise gives that K (min tan γ ) 1, where i1 1 ≤ ı ı − tan γı = dı/bı. We need a lower bound for tan γı. From (2.1.11) we get that

n ` n ` di1...in 1 a 1 da a tan γi1...in = = ∑ di` = + ∑ di` . bi1...in a b a b b ! `=1 `=2 This is minimal if d = d for every ` 2. Thus, we obtain the lower bound i` − ≥

n ` 1 d a − d a/b d(b 2a) tan γi1...in 1 ∑ 1 = − . ≥ b − b ! ≥ b − 1 a/b b(b a) `=2 − − This remains positive iff a < b/2 = 1/6. Substituting d and b gives the bound for K1. Corollary 2.7.2. For every a < 1/6 : dim Λ = dim Λ = 1 log 2/ log a. H a B a − Proof. For uniform vertical ﬁbres both conditions (2.2.2) and (2.2.5) simplify to

log a log N > , which is satisﬁed here iff a (0, 1/6). log b log M ∈ 2.7. Examples 55

Thus, Claim 2.7.1 and Corollary 2.2.12 together imply that for every a (0, 1/6) we ∈ have dim Λ = dim Λ = 1 log 2/ log a. H a B a − 2.7.4 Example "X X" ≡ This diagonally homogeneous carpet, recall Figure 1.10, is a modiﬁcation of the previous from Subsection 2.7.3 in order to show an overlapping example for which all dimensions are different. Indeed, clearly it does not have uniform vertical ﬁbres and there are empty columns. The main diagonal of each matrix in the TGL IFS is b b = 0.28 and a a. The i ≡ i ≡ off-diagonal elements are either d = (1/2 a) or 0. The translation vectors were i ± − chosen so that Λa is symmetric on both lines x = 1/2 and y = 1/2. In Figure 1.10 a = 0.045. Transversality for the system can be checked the same way as in Claim 2.7.1, to obtain that transversality holds for every a < b/2 = 0.14 with

0.28(0.28 a) K < − . 1 (1/2 a)(0.28 2a) − −

Corollary 2.7.3. We have dimH Λa < dimB Λa < dimAff Λa, where

1.27297 1.27297 dim Λ = 0.78556 log 2 2 log a + 3 log a , for every a < 0.10405 . . . , H a · · − − 0.84730 dim Λ = + 0.86303, for every a < 0.10254 . . . , B a log a − 0.67294 dim Λ = 1 + , for every a < 0.28 . Aff a log a − Proof. The formulas are applications of the ones in Corollary 2.2.12 and (2.2.3). The afﬁnity dimension is independent of overlaps. The bound for a in case of the Haus- dorff dimension was obtained using Proposition 2.2.10. The value x0 = 0.56255 . . . for which R(x0) = 1 was calculated using Wolfram Mathematica 11.2. Then (2.2.2) holds for every a < b1/x0 = 0.10405 . . . . The bound on a for the box dimension simply comes from substituting the parameters into the second inequality in (2.2.9).

2.7.5 Negative entries in the main diagonal

Throughout we assumed that 0 < ai < bi < 1. We now comment on letting ai or bi < 0. For convenience, assume ROSC and non-overlapping columns. Proposition 2.7.4. The dimension results of Theorems 2.2.2 and 2.2.4 extend to TGL carpets satisfying the ROSC under the weaker condition that 0 < a < b < 1 and for every ﬁxed | i| | i| ıˆ 1, . . . , M and every k, ` : b = b . ∈ { } ∈ Iıˆ k ` Sketch of proof. All lower triangular matrix can be written

b 0 b 0 i = | i| L di ai d¯i ai · | | where d¯ = d or d and L is a reﬂection on one or both of the coordinate axis. Since i i − i L([ 1, 1]2) = [ 1, 1]2, such compositions ﬁt into the framework of Fraser’s box-like − − sets [Fra12]. Furthermore, the direction-x dominates property is preserved. Hence, the proof of the box dimension from Section 2.6 immediately extends to this setting. 56 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

The lower bound for the Hausdorff dimension follows from Bárány–Käenmäki [BK17] cited in Theorem 2.3.4. Since in any given column all bi have the same sign and we have ROSC, the column structure is preserved for every level. Thus, the dimension of the projected measure νq is not affected by the negative ai, bi. For the upper bound, we can modify the metric deﬁned on Σ in Lemma 2.4.1 to be

ıˆ ˆ i j | ∧ | | ∧ | d( ) = b + a i, j : ∏ ik ∏ ik . k=1 | | k=1 | |

One can easily check that d(i, j) is indeed a metric and the natural projection Π : Σ → Λ is Lipschitz. Only the lengths of the sides of a parallelogram are important, its orientation is not. The Bernoulli measure deﬁned in (2.4.3) can be modiﬁed by again putting ai and bi in absolute value. The original proof of Gatzouras and Lalley [GL92] does not use that a , b > 0, only that 0 < a < b < 1. i i | i| | i| In general, if a column has bi of different signs, then the initial column structure can easily be destroyed. This is true even if b b and possibly empty columns | i| ≡ also have width b, see Figure 2.5. This motivates us to call a TGL carpet symmetric if Nıˆ = NM ıˆ+1 for ıˆ = 1, . . . , M/2 (empty columns are allowed) and bi 1/M. − b c | | ≡ For a particular symmetric carpet, in the next subsection, we show that the dimension formulas hold.

FIGURE 2.5: Orientation reversing maps generally destroy the column structure. First and second level cylinders of the horizontal IFS are shown, arrows indicating the orientation. Left: different b ,H right: | i| equal b and gap as well. e | i|

2.7.6 A family of self-afﬁne continuous curves Let a (0, 1/5] and d = (1 5a)/4. Deﬁne the matrices ∈ − 1/3 0 1/3 0 A = and A− = − . d a 0 a A is orientation reversing. We introduce the parameterized family of IFSs given − Fa by the functions

1/3 2/3 f (x) = Ax, f (x) = Ax + , f (x) = A x + , 1 2 a + d 3 − 2(a + d) 1/3 2/3 f (x) = Ax + , f (x) = Ax + . 4 3a + 2d 5 4a + 3d

The translation vectors are chosen so that f1(0) = 0, f5((1, 1)) = (1, 1) and fi((1, 1)) = 2 fi+1(0). This ensures that Λa is a continuous curve in R , see Figure 1.11. Curves satisfying this property are also called afﬁne zippers in the literature, see for example [ATK03; BKK18]. Clearly, the attractor Λa is a symmetric, diagonally homogeneous TGL carpet satisfying the ROSC for every value of a. For a = 1/5 all cylinders Ri n are | 2.7. Examples 57

rectangles, however it is not a classical Bedford-McMullen carpet, since A− contains a negative element.

Proposition 2.7.5. For every a (0, 1/5], the Hausdorff and box dimension of Λ are given ∈ a by the continuous, strictly increasing functions

1 log 3 log(3/5) log 2 + 3 log a = dim Λ < dim Λ = 1 + . log 3 · − H a B a log a Proof.A − can be written as the composition of the reflection on the vertical axis with the diagonal matrix Diag(1/3, a). Hence, the proof of the box dimension carries over without difficulty. The argument for the Hausdorff dimension follows that in Proposition 2.7.4, with an extra argument why the dimension of νq is not affected by A−. The symbolic space Σ = 1, . . . , 5 N codes the IFS on [0, 1]2 and on [0, 1] { } Fa Ha (recall (2.1.2)). Fix a p = (p ,..., p ) . Due to the symmetry and diagonally 1 5 ∈ P homogeneous property we may assume that p1 = p5. Let µp be the Bernoullie measure on Σ and νp = Π µp its push forward. Define the IFS a := hi(x) = x/3 + (i ∗ N H { − 1)/3, i = 1, 2, 3 , which is coded by Σ = 10, 20, 30 . The map φ : 1, . . . , 5 } H { } { } → 1 , 2 , 3 is defined { 0 0 0}

φ(1) = 10, φ(2) = φ(3) = φ(4) = 20, φ(5) = 30.

For ı = i ... i 1 , 2 , 3 n let us denote Jk(ı) := j : i = k, j ı ,#k(ı) := Jk(ı) 1 n ∈ { 0 0 0} { j ≤ | |} | | and deﬁne νq := (proj ) νp. We claim that x ∗

10 ( )+ 30 ( ) 2 ν (h ([0, 1])) = p# ı # ı (p + p + p )# 0 (ı), (2.7.2) q ı 1 · 2 3 4 i.e. νq is the push forward (Π ) µq of the Bernoulli measure µq on Σ deﬁned by the H ∗ H vector q = (q1, q2, q3) = (p1, p2 + p3 + p4, p5). This implies that

log q q dim ν = h i . H q log 3 − To see (2.7.2), choose an arbitrary ı 1 , 2 , 3 , n. We determine those ıˆ 1, . . . , 5 n ∈ { 0 0 0 } ∈ { } for which proj f ([0, 1]2) = h ([0, 1]). For indices j J20 (ı) we can choose 2, 3 or 4 in ıˆ. x ıˆ ı ∈ Let J3(ıˆ) := j : ıˆ = 3, j ` ıˆ J20 (ı). Orientation is reversed at each j J3(ıˆ). ` { j ≤ ≤ | |} ⊆ ∈ ` J3(ıˆ) uniquely determines ıˆ if i = 1 or 3 . Namely, whenever | ` | ` ` 0 0

3 odd, if i` = 10 then necessarily ıˆ` = 5 and if i` = 30 then ıˆ` = 1; J` (ıˆ) is | | (even, if i` = 10 then necessarily ıˆ` = 1 and if i` = 30 then ıˆ` = 5.

20 3 For indices j J (ı) J ıˆ (ıˆ) we can freely choose ıˆj = 2 or 4. These are precisely the ıˆ ∈ \ 2| | for which projx fıˆ([0, 1] ) = hı([0, 1]). Using that p1 = p5, the measure equals

2 2 3 1(ı)+ 5(ı) # 0 (ı) 3(ı) # 0 (ı) # (ıˆ) 2(ı) 4(ı) ν (h ([0, 1])) = p# ˆ # ˆ p# ˆ − p# ˆ p# ˆ , q ı 1 #3(ıˆ) 3 #2(ıˆ) 2 · 4 which after two applications of the binomial theorem yields (2.7.2). Finally, we conclude that dimH Λa < dimB Λa since Λa does not have uniform vertical ﬁbres. 58 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

2.8 Three-dimensional applications

We can compute the Hausdorff dimension of some self-affine carpets in R3. We do not aim for full generality, rather just demonstrate how our results can be applied. Throughout this section we always use the following definitions: Definition 2.8.1. Let be a TGL carpet on [0, 1]2 of the form (2.1.1), that is F

N bi 0 ti,1 2 = fi(x) := Ai x + ti i=1, where Ai = and ti = , x [0, 1] . F { · } di ai ti ∈ ,2

Furthermore, let the vectors u = (u1,..., uN), v = (v1,..., vN), λ = (λ1,..., pλN) be such that for every 1 i N ≤ ≤ u , v R and λ ( 1, 1) 0 . i i ∈ i ∈ − \ { } We say that the three dimensional self-afﬁne IFS

N bi 0 0 ti,1 := Fi(x) := Ai x + ti , where Ai = di ai 0 , ti := ti,2 F · i=1     n o ui vi λi ti,3 b b b b b b   b   on [0, 1]3 is an uplift of corresponding to (u, v, λ) if the following conditions hold: F (C1) For all 1 i N we have ≤ ≤ 0 < λ < a < b < 1. (2.8.1) | i| i i (C2) satisfies the ROSC (see Definition 2.1.3). F Let Λ and Λ be the attractor of and respectively. We write Π and Π for the natural b F F projection from Σ := 1, . . . , N N to Λ and Λ respectively. For a probability vector p := { }N N (p1,..., pN) web set νp := Π (p ) and νp b:= Π (p ) b ∗ b ∗ We obtain as a corollary of [BK17, Theorem 2.3, Proposition 5.8 and Proposition b 5.9] that b Corollary 2.8.2 (Bárány, Käenmäki). Assume that for an uplift of we have u = v = 0 F F and all components of λ are equal to the same λ. Moreover, assume that for a probability 1 2 vector p = (p1,..., pN) we have hp < χp + χp (i.e. the entropyb is less than the sum of the Lyapunov exponents). Then dimH νp = dimH νp. That is, the computation of the Hausdorff dimension of a Bernoulli measure for the three-dimensional non-overlapping systemb is traced back to the corresponding F two-dimensional possibly overlapping system . In this way, if satisfies the con- F F ditions of Theorem 2.2.2 then we can determineb dimH(νp) for the three-dimensional system. In general, we cannot approximate the Hausdorff dimensionb of a self-affine set in R3 by the Hausdorff dimension of self-affine (or even ergodic) measures (see [DS17, Theorem 2.8]). However, this is possible in some special cases. Theorem 2.8.3. Given a diagonally homogeneous TGL of the form

N b 0 ti,1 2 = fi(x) := A x + ti i=1, where A = and ti = , x [0, 1] , F { · } di a ti ∈ ,2 we assume that 2.8. Three-dimensional applications 59

(i) has uniform vertical ﬁbres (i.e. each column has the same number of maps). F (ii) The projection of Λ to the x-axis is the whole interval [0, 1] (this means that 1/b is equal to the number of columns M). We assume this to guarantee that the box and afﬁnity dimensions of Λ coincide (see Corollary 2.2.5).

(iii) Moreover, we assume that the parameter a is sufﬁciently small so that both conditions (2.2.10), (2.1.16) and the transversality condition hold:

log N bd a < min b log M , ∗ , (2.8.2) 2 + d ∗ where d was deﬁned in Lemma 2.1.8 as ∗

d := min min dk d` , 1 ˆ M ∗ (k,`) ˆ | − | ≤ =≤∅ Pˆ6 ∈P

where (k, `) if f ([0, 1]2) and f ([0, 1]2) belong to the same column and have ∈ Pˆ k ` disjoint interior.

We consider the self-affine IFS which is an uplift of corresponding to (u, v, λ) according F F to Definition 2.8.1. That is (2.8.1) holds and u, v and λ are chosen such that b b 0 0 ti,1 N := F (x) := A x + t , where A = d a 0 , t := t , x [0, 1]3 F i i · i i=1 i  i  i  i,2 ∈ u v λ t i i i i,3 b b b b b b   b   b satisfies:

F [0, 1]3 [0, 1]3 holds for all i 1, . . . , N and • i ⊂ ∈ { } the set F [0, 1]3 F [0, 1]3 has empty interior for all i = j 1, . . . , N . • i ∩ j 6 ∈ { } Let p := 1/N, . . . , 1/N . Using the notation of Deﬁnition 2.8.1 we have N | {z } log(Nb) dim ν = dim Λ = dim Λ = dim Λ = 1 + . (2.8.3) H p H B Aff log a − To give the upperb bound inb the proof ofb this theorem,b ﬁrst we need to extend the scope of Lemma 2.1.5 to R3.

Lemma 2.8.4. There exists K , K and K such that for an arbitrary n and (i ,..., i ) x y z 1 n ∈ (1, . . . , N)n we have bn 0 0 A K bn an 0 , i1...in ≤  x ·  K bn K bn λ y · z · i1...in b   that is all the elements of the matrix on the right-hand side are greater than or equal to the corresponding element on the left-hand side.

Proof. For every n and (i ,..., i ) (1, . . . , N)n we introduce x , y and z 1 n ∈ i1...in i1...in i1...in such that bn 0 0 A = x bn an 0 . i1...in  i1...in ·  y bn z bn λ i1...in · i1...in · i1...in b   60 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

Since the existence of Kx was proved in Lemma 2.1.5, it sufﬁces to prove that yi1...in and z are uniformly bounded in (i ,..., i ) Σ . To do so, observe that i1...in 1 n ∈ ∗ a λ vi z = z + i1...in n+1 , (2.8.4) i1...in+1 i1...in b bn b di λ ui y = y + z n+1 + i1...in n+1 . (2.8.5) i1...in+1 i1...in i1...in b bn b By (2.8.1) we obtain from (2.8.4) that there is an r (0, 1) and c > 0 such that ∈ z < c rn for all n and (i ,..., i ) 1, . . . , N n . (2.8.6) i1...in · 1 n ∈ { }

Namely, we can write down the formula for zi1...in inductively and thus we get that n max v z (a/b) max v + n i{ i} . From here we get that (2.8.6) holds. This i1...in ≤ · i i b settles the existence of nKz. Substitutingo (2.8.6) into (2.8.5) and using (2.8.1) again we obtain the existence of Ky. Namely, the second and third summands in (2.8.5) are exponentially small. More precisely,

∞ max d max λ n max u K = max u + c rn { i} + {| i|} { i} , y { i} ∑ · · b b · b n=1 where all of the maximums are taken for i 1, . . . , N . ∈ { } Proof of Theorem 2.8.3.

Lower bound Observe that if condition (2.8.2) holds then it follows from Lemma 2.1.8 that the transversality condition holds. Moreover, as we noted in Section 2.2.3, condition (2.8.2) also implies that conditions (2.2.2) and (2.2.5) hold when p is chosen as above to be the uniform vector. In this way the conditions of Theo- rems 2.2.2 and 2.2.7 are satisﬁed. As an application of these theorems, we obtain that log N log b log M log(Nb) dim ν = + 1 = 1 + . H p log a − log a log b log a − − − This implies that

log(Nb) 1 + < dim ν dim ν dim Λ. log a H p ≤ H p ≤ H − b b Upper bound It is enough to prove that

log(Nb) dim Λ 1 + . (2.8.7) Aff ≤ log a − b 3 This follows from Lemma 2.8.4 since the cylinder Fi1,...in [0, 1] can be covered by Nn bn/an axes parallel rectangular box of dimensions an K an (K + · × x · × y K ) an. This immediately implies that (2.8.7) holds. z · 2.8. Three-dimensional applications 61

Example 1. Recall the attractor in the center of Figure 1.12. It is deﬁned by an IFS

6 = F (x) := A x + t , where for 0 < λ < a < 1/3 F i i · i i=1 1/3 0 0 1/3 0 0 1/3 0 0 Ab = A =b 1 bab a b0 , A = A = a 1 a 0 , A = A = 0 a 0 . 1 5  −  2 6  −  3 4   1 λ 0 λ 0 0 λ λ 1 0 λ − − b b   b b   b b   The translations are chosen appropriately so that satisﬁes the ROSC and the projection to F the xy-plane looks like the one on the right-hand side of Figure 1.12. If λ < a < 1/6, then the conditions of Theorem 2.8.3 hold and we have fromb (2.8.3) that for p = (1/6, . . . , 1/6)

log 2 dim ν = dim Λ = dim Λ = dim Λ = 1 . H p H B Aff − log a b b b b Open problems We plan to study the appropriate dimensional Hausdorff measure of planar carpets. This is always more difficult than determining the actual dimension. Determining whether it is 0, infinite or positive and finite is the goal. Not much is known about this for planar carpets. It is known for Gatzouras–Lalley carpets that if the Hausdorff and box dimension are equal, then the Hausdorff measure is positive and finite [GL92]. Moreover, if they are not equal, then Peres showed for Bedford–McMullen carpets that the Hausdorff measure is infinite and the set is not σ-finite with respect to the Hausdorff measure. Natural questions can be the following

1. Can the proof of Peres be adapted in order to extend the result to the more general (triangular) Gatzouras–Lalley case?

2. When the measure is infinite, we do not expect overlaps to cause the value of the measure to drop to a finite value. However, when the Hausdorff and box dimension are equal, and thus the measure is positive and finite, then can overlaps cause the Hausdorff measure to drop to zero? A similar phenomena was proved for solenoids by Rams and Simon [RS03]. We plan to first look at simple examples and then see to what extent can those findings be generalized.

Chapter 3

Pointwise regularity of parameterized afﬁne zipper fractal curves

This chapter is based on the article [BKK18] written jointly with Balázs Bárány and Gergely Kiss.

3.1 Self-afﬁne zippers satisfying dominated splitting

Let us begin by deﬁning fractal curves generated by zippers. To the best of our knowledge, the terminology and deﬁnition of a zipper in this generality is due to Aseev, Tetenov and Kravchenko [ATK03]. However, special cases already appear in the works of Hutchinson [Hut81] and Barnsley [Bar86].

d Deﬁnition 3.1.1. A system = f0,..., fN 1 of contracting mappings of R to itself F { − } is called a zipper with vertices Z = z0,..., zN and signature ε = (ε0,..., εN 1), εi { } − ∈ 0, 1 , if the cross-condition { }

fi(z0) = zi+ε and fi(zN) = zi+1 ε i − i holds for every i = 0, . . . , N 1. We call the system a self-afﬁne zipper if the functions f − i are afﬁne contractive mappings of the form

f (x) = A x + t , for every i 0, 1, . . . , N 1 , i i i ∈ { − } where A Rd d invertible and t Rd. i ∈ × i ∈ The fractal curve generated from is the unique non-empty compact set Γ, for which F N 1 − Γ = fi(Γ). i[=0 If is an affine zipper then we call Γ a self-affine curve. F For an illustration see Figure 3.1. It shows the first (red), second (green) and third (black) level cylinders of the image of [0, 1]2. The cross-condition ensures that Γ is a continuous curve. The dimension theory of self-affine curves is far from being well understood. The Hausdorff dimension of such curves is known only in a very few cases. The 64 Chapter 3. Pointwise regularity of parameterized affine zipper fractal curves

FIGURE 3.1: An affine zipper with N = 3 maps and signature ε = (0, 1, 0). usual techniques, like self-affine transversality, see Falconer [Fal88b], Jordan, Polli- cott and Simon [JPS07], destroys the curve structure. Ledrappier [Led92] gave a sufficient condition to calculate the Hausdorff dimension of some fractal curves, and Solomyak [Sol98b] applied it to calculate the dimension of the graph of the Takagi function for typical parameters. Feng and Käenmäki [FK18] characterized self-affine systems, which have analytic curve attractor. Bandt and Kravchenko [BK11] studied some smoothness properties of self-affine curves, especially the tangent lines of planar self-affine curves. In Subsection 2.7.6 we used the framework of the TGL carpets of Chapter2 to calculate the different values of the Hausdorff and box dimension of a whole family of self-affine zippers. In this chapter we use the same notation as in Chapter2 for the various dimensions of sets and measures. Let us recall the definition of pointwise Hölder exponent of a real valued function g, see for example [Jaf97a, eq. (1.1)]. We say that g Cβ(x) ∈ if there exist a δ > 0, C > 0 and a polynomial P with degree at most β such that b c g(y) P(y x) C x y β for every y B (x), | − − | ≤ | − | ∈ δ where B (x) denotes the ball with radius δ centered at x. Let α (x) = sup β : g δ p { ∈ Cβ(x) . We call α (x) the pointwise Hölder exponent of g at the point x. } p In this chapter, we study the local regularity of a generalized version of self-similar functions F, recall (1.4.1). Namely, let λ = (λ0,..., λN 1) be a probability vector. Let − us subdivide the interval [0, 1] according to the probability vector λ and signature ε = (ε0,..., εN 1), εi 0, 1 of the zipper . Let gi be the affine function mapping − ∈ { } F the unit interval [0,1] to the ith subinterval of the division which is order-preserving or order-reversing according to the signature εi. That is, the interval [0, 1] is the attractor of the iterated function system

ε N 1 = g : x ( 1) i λ x + γ − , (3.1.1) G { i 7→ − i i}i=0 i 1 N 1 d+1 where γ = ∑ − λ + ε λ . Let = S − be an IFS on R such that i j=0 j i i S { i}i=0

Si(x, y) = (gi(x), Aiy + ti).

It is easy to see that if Λ is the attractor of then for every x [0, 1] there exists a S ∈ unique y Rd such that (x, y) Λ. Thus, we can define a function v : [0, 1] Γ Rd ∈ ∈ 7→ ⊂ 3.1. Self-affine zippers satisfying dominated splitting 65 such that v(x) = y if (x, y) Λ. The function v satisfies the functional equation ∈ 1 v(x) = f v(g− (x)) if x g ([0, 1]). (3.1.2) i i ∈ i 1 x γi 1 We note that g− (x) = −ε , and g− (x) [0, 1] if and only if x gi([0, 1]). More- i ( 1) i λi i ∈ ∈ N 1 − over, if f − is a self-affine zipper then v is continuous. We call v as the linear { i}i=0 parametrization of Γ. Such a particular example is de Rham’s curve, introduced in (1.2.3), see also Section 3.6 for details. As a slight abuse of the appellation of the pointwise Hölder exponent, we use another exponent α(x) of the function v at a point x [0, 1] ∈ log v(x) v(y) α(x) = lim inf k − k. (3.1.3) y x log x y → | − |

We note that if αp(x) < 1 or α(x) < 1 then αp(x) = α(x). Otherwise, we have only α(x) α (x). ≤ p When the lim inf in (3.1.3) exists as a limit, then we say that v has a regular pointwise Hölder exponent α (x) at a point x [0, 1], i.e. r ∈ log v(x) v(y) αr(x) = lim k − k. (3.1.4) y x log x y → | − | Let us deﬁne the level sets of the (regular) pointwise Hölder exponent by

E(β) = x [0, 1] : α(x) = β and E (β) = x [0, 1] : α (x) = β . (3.1.5) { ∈ } r { ∈ r } Our goal is to perform multifractal analysis, i.e. to study the maps

β dim E(β) and β dim E (β). 7→ H 7→ H r

Dominated splitting

o d 1 Let us denote by M the interior and by M the closure of a set M PR − . For a d ⊆ d 1 point v R , denote by v the equivalence class of v in the projective space PR − . ∈ h i d 1 Every invertible matrix A defines a natural map on the projective space PR − by v Av . As a slight abuse of notation, we denote this function by A too. h i 7→ h i Definition 3.1.2. We say that a family of matrices = A0,..., AN 1 have dominated A { − } d 1 splitting of index-1 if there exists a non-empty open subset M PR − with a finite ⊂ number of connected components with pairwise disjoint closure such that

N 1 − A M Mo, i ⊂ i[=0 and there is a d 1 dimensional hyperplane that is transverse to all elements of M. We call − the set M a multicone.

We adapted the definition of dominated splitting of index-1 from the paper of Bochi and Gourmelon [BG09]. They showed that the tuple of matrices satisfies the A property in Definition 3.1.2 if and only if there exist constants C > 0 and 0 < τ < 1 such that α (A A ) 2 i1 ··· in Cτn α (A A ) ≤ 1 i1 ··· in 66 Chapter 3. Pointwise regularity of parameterized affine zipper fractal curves

for every n 1 and i0,..., in 1 0, . . . , N 1 , where αi(A) denotes the ith largest ≥ − ∈ { − } singular value of the matrix A. That is, the weakest contracting direction and the stronger contracting directions are strongly separated away (splitted), α1 dominates α2. This condition makes it easier to handle the growth rate of the norm of matrix products, which will be essential in our later studies. We note that for example a tuple formed by matrices with strictly positive el- A ements, satisfies the dominated splitting of index-1 of M = x Rd : x > 0, i = h{ ∈ i 1, . . . , d . Throughout the paper we work with affine zippers, where we assume that }i the matrices Ai have dominated splitting of index-1. For more details, see Section 3.3 and [BG09]. d 1 d For a subset M of PR − and a point x R , let ∈ M(x) = y Rd : y x M . (3.1.6) { ∈ h − i ∈ } We call the set M(x) a cone centered at x. We call a non-degenerate system, if it satisfies the SOSC, recall Definition 1.2.1 and F ∞ 1 c z z / A− (M ), (3.1.7) h N − 0i ∈ ı k=0 ı =k \ |\| where for a finite length word ı = i1 ... ik, Aı denotes the matrix product Ai1 Ai2 ... Aik . We note that the non-degenerate condition guarantees that the curve v : [0, 1] Rd 7→ is not self-intersecting and it is not contained in a strict hyperplane of Rd.

3.2 Main results

The key technical tool for our work is the matrix pressure function. Denote by P(t) the pressure function which is deﬁned as the unique root of the equation

N 1 1 − t P(t) 0 = lim log Ai Ai (λi λi )− . (3.2.1) n ∞ n ∑ k 1 ··· n k 1 ··· n → i1,...,in=0

A considerable attention has been paid for pressures, which are deﬁned by matrix norms, see for example Käenmäki [K04¨ ], Feng and Shmerkin [FS14], and Morris [Mor16; Mor17]. Feng [Fen03] and later Feng and Lau [FL02] studied the properties of the pressure P for positive and non-negative matrices. In Section 3.3, we extend these results for the dominated splitting of index-1 case. Namely, we will show that the function P : R R is continuous, concave, monotone increasing, and continuously 7→ differentiable. Let d0 > 0 be the unique real number such that P(d0) = 0, i.e.

1 d0 0 = lim log Aı . (3.2.2) n ∞ n ∑ → ı =n k k | | Observe that for every n 1, f (U) : ı = n deﬁnes a cover of Γ. But since Γ is ≥ { ı | | } a curve and thus dim Γ 1, and since every f (U) can be covered by a ball with H ≥ ı radius A U we have d 1. k ık| | 0 ≥ Let P(t) P(t) αmin = lim , αmax = lim and α = P0(0). (3.2.3) t +∞ t t ∞ t → →− b 3.2. Main results 67

The values αmin and αmax correspond to the logarithm of the joint- and the lower- spectral radius defined by Protasov [Pro06]. Now, we state our main theorems on the pointwise Hölder exponents. Theorem 3.2.1. Let v be a linear parametrization of Γ defined by a non-degenerate system . F Then there exists a constant α, defined in (3.2.3), such that for -a.e. x [0, 1], α(x) = α L ∈ ≥ 1/d . In addition, there exists an ε > 0 such that for every β [α, α + ε] 0 ∈ b b dimH x [0, 1] : α(x) = β = inf tβ P(t) . (3.2.4) { ∈ } t R{ −b b } ∈ Moreover, (3.2.4) can be extended for every β [α , α + ε] if v satisfies ∈ min k A0 b λ0 = λN 1 and lim k k = 1. (3.2.5) − k ∞ Ak → N 1 k − k Furthermore, the functions β dim E(β) and β dim E (β) are continuous and 7→ H 7→ H r concave on their respective domains.

In the following, we give a sufficient condition to extend the previous result, where (3.2.4) holds to the complete spectrum [αmin, αmax]. As a slight abuse of no- d 1 tation for every θ PR − , we say that 0 = v θ if v = θ. ∈ 6 ∈ h i N 1 Assumption A. For a non-degenerate affine zipper = f : x A x + t − with F { i 7→ i i}i=0 vertices z0,..., zN assume that there exists a convex, simply connected closed cone C d 1 { } ⊂ PR − such that 1. N A C Co and for every 0 = v θ C, A v, v > 0, i=1 i ⊂ 6 ∈ ∈ h i i 2. Sz z Co. h N − 0i ∈ Observe that if satisfies Assumption A then it satisfies the strong open set con- F dition with respect to the set U, which is the bounded component of Co(z ) Co(z ). 0 ∩ N We note that if all the matrices have strictly positive elements and the zipper has signature (0, . . . , 0) then Assumption A holds. Theorem 3.2.2. Let be an affine zipper satisfying Assumption A. Then for every β F ∈ [α, αmax] dimH x [0, 1] : α(x) = β = inf tβ P(t) , (3.2.6) { ∈ } t R{ − } b ∈ and for every β [α , α ] ∈ min max

dimH x [0, 1] : αr(x) = β = inf tβ P(t) . (3.2.7) { ∈ } t R{ − } ∈ Moreover, if satisﬁes (3.2.5) then (3.2.6) can be extended for every β [α , α ]. F ∈ min max The functions β dim E(β) and β dim E (β) are continuous and concave on 7→ H 7→ H r their respective domains.

Assumption A has another important role. In Theorem 3.2.2, we calculated the spectrum for the regular Hölder exponent, providing that it exists. We show that the existence of the regular Hölder exponent for Lebesgue typical points is equivalent to Assumption A. Theorem 3.2.3. Let be a non degenerate system. Then the regular Hölder exponent exists F for Lebesgue almost every point if and only if satisﬁes Assumption A. In particular, α (x) = F r P (0) for Lebesgue almost every x [0, 1]. 0 ∈ 68 Chapter 3. Pointwise regularity of parameterized afﬁne zipper fractal curves

Remark 3.2.4. In the sequel, to keep the notation tractable we assume the signature ε = (0, . . . , 0). The results carry over for general signatures, and the proofs can be easily modiﬁed for the general signature case, see Remark 3.6.3.

The organization of the chapter is as follows. In Section 3.3 we prove several properties of the pressure function P(t), extending the works of [Fen03; FL02] to the dominated splitting of index-1 case using [BG09]. We prove Theorem 3.2.1 in Sec- tion 3.4. Section 3.5 contains the proofs of Theorems 3.2.2 and 3.2.3 when the zipper satisﬁes Assumption A. Finally, as an application in Section 3.6, we show that our results can be applied to de Rham’s curve, giving ﬁner results than existing ones in the literature.

3.3 Pressure for matrices with dominated splitting of index-1

In this section, we generalize the result of Feng [Fen03], and Feng and Lau [FL02]. In [FL02] the authors studied the pressure function and multifractal properties of Lya- punov exponents for products of positive matrices. Here, we extend their results for a more general class of matrices by using Bochi and Gourmelon [BG09] for later usage. Let Σ be the set of one side infinite length words of symbols 0, . . . , N 1 , i.e. N { − } Σ = 0, . . . , N 1 . Let σ denote the left shift on Σ, its n-fold composition by { − } σni = (i , i ,...). We use the standard notation i n for i ,..., i and n+1 n+2 | 1 n [i ] := j Σ : j = i ,..., j = i . |n { ∈ 1 1 n n} Let us denote the set of finite length words by Σ = ∞ 0, . . . , N 1 n, and for ∗ n=0{ − } an ı Σ , let us denote the length of ı by ı . For a finite word ı Σ and for a j Σ, ∈ ∗ | | S ∈ ∗ ∈ denote ıj the concatenation of the finite word ı with j. Denote i j the length of the longest common prefix of i, j Σ, i.e. i j = ∧ ∈ ∧ min n 1 : in = jn . Let λ = (λ0,..., λN 1) be a probability vector and let d(i, j) be { − 6 } − the distance on Σ with respect to λ. Namely,

i j ∧ d(i j) = λ = λ , ∏ in : i i j . (3.3.1) n=1 | ∧

If i j = 0 then by deﬁnition i i j = ∅ and λ∅ = 1. In the sequel, whenever we use ∧ | ∧ Hausdorff dimension in Σ it is with respect to this metric d(i, j). For every r > 0, we deﬁne a partition Ξr of Σ by

Ξr = [i1,..., in] : λi1 λin r < λi1 λin 1 . (3.3.2) ··· ≤ ··· − For a matrix A and a subspace θ, denote A θ the norm of A restricted to θ, i.e. k | k A θ = sup Av / v . In particular, if θ has dimension one A θ = Av / v k | k v θ k k k k k | k k k k k for any 0 = v∈ θ. Denote G(d, k) the Grassmanian manifold of k dimensional sub- 6 ∈ spaces of Rd. We deﬁne the angle between a 1 dimensional subspace E and a d 1 − dimensional subspace F as usual, i.e.

v, proj v (E, F) = arccos h F i , ^ proj v v k F kk k where 0 = v E arbitrary and proj denotes the orthogonal projection onto F. 6 ∈ F The following theorem collects the most relevant properties of a family of matrices with dominated splitting of index-1. 3.3. Pressure for matrices with dominated splitting of index-1 69

Theorem 3.3.1. [BG09, Theorem A, Theorem B, Claim on p. 228] Suppose that a finite set of matrices A0,..., AN 1 satisfies the dominated splitting of index-1 with multicone M. { − } d 1 Then there exist Hölder continuous functions E : Σ PR − and F : Σ G(d, d 1) 7→ 7→ − such that 1.E (i) = A E(σi) for every i Σ, i1 ∈ 1 2.F (i) = A− F(σi) for every i Σ, i1 ∈ 3. there exists β > 0 such that (E(i), F(j)) > β for every i, j Σ, ^ ∈ 4. there exist constants C 1 and 0 < τ < 1 such that ≥ α2(Ai ) |n Cτn Ai ≤ k |n k for every i Σ and n 1, ∈ ≥ n 5. there exists a constant C > 0 such that Ai E(σ i) C Ai for every i Σ, k |n | k ≥ k |n k ∈ 6. there exists a constant C > 0 such that Ai F(in ... i1j) Cα2(Ai ) for every k |n | k ≤ |n i, j Σ. ∈ There are a few simple consequences of Theorem 3.3.1. First, if M is the multicone from Definition 3.1.2, then by Theorem 3.3.1(1)

∞ E(i) = A A (M), i1 ··· in n\=1 and for every V M, A A V E(i) uniformly (independently of V). Hence, ∈ i1 ··· in → by property (5), there exists a constant C > 0 such that for every V M and every 0 ∈ ı Σ , ∈ ∗ A V C0 A . (3.3.3) k ı| k ≥ k ık So, this gives us a strong control over the growth rate of matrix products on subspaces in M. Remark 3.3.2. We note if the multicone M in Deﬁnition 3.1.2 has only one connected component then it can be chosen to be simply connected and convex. Indeed, since M is separated away from the strong stable subspaces F then cv(M) must be separated away from every d 1 − dimensional strong stable subspace, as well, where cv(M) denotes the convex hull of M. Thus A (cv(M)) cv(M)o for every i. i ⊂ Second, property (1) of Theorem 3.3.1 also implies that

n A E( ni) = A E( ki) i n σ ∏ ik σ . (3.3.4) k | | k k=1 k | k

Indeed, since E(i) is a one dimensional subspace, for every v E(σni) ∈ n n Ai v Ai ...i v n |n k n k Ai E(σ i) = k k = k k = Aik E(σ i) . k |n | k v ∏ A v ∏ k | k k k k=1 k ik+1,...,in k k=1 Moreover, since E(i) is Hölder-continuous, the function ψ(i) := log A E(σi) is k i1 | k also Hölder-continuous. That is, there exist C > 0 and 0 < τ < 1 such that

i j ψ(i) ψ(j) Cτ ∧ . (3.3.5) | − | ≤ 70 Chapter 3. Pointwise regularity of parameterized afﬁne zipper fractal curves

It is easy to see that (3.3.5) holds if and only if ψ is Hölder-continuous with respect to the metric d defined in (3.3.1). Finally, (3.3.5) implies that for every t, the potential function ϕ : Σ R defined t 7→ by t P(t) ϕt(i) := log Ai E(σi) λ− = tψ(i) P(t) log λi (3.3.6) k 1 | k i1 − 1 is Hölder-continuous w.r.t the metric d, where P(t) was defined in (3.2.1). Thus, by [Bow08, Theorem 1.4], for every t R there exists a unique σ-invariant, ergodic ∈ probability measure µt on Σ such that there exists a constant C(t) > 1 such that for every i Σ and every n 1 ∈ ≥

1 µt([i n]) µt([i n]) C(t)− | = | C(t), (3.3.7) n 1 ϕ (σki) P(t) ≤ ∏ − e t n t − ≤ k=0 Ai n E(σ i) λi k | | k · |n where the equality follows from substituting (3.3.6) and (3.3.4). Moreover, for the Hausdorff dimension w.r.t. the metric d deﬁned in (3.3.1)

hµt dimH µt = , (3.3.8) χµt where 1 hµ = lim − µt([ı]) log µt([ı]) = ϕt(i)dµt(i), (3.3.9) t n ∞ n ∑ → ı =n − Z | | 1 χµ = lim − µt([ı]) log λı = log λi dµt(i). (3.3.10) t n ∞ n ∑ 1 → ı =n − Z | |

We call χµt the Lyapunov exponent of µt and hµt the entropy of µt. Lemma 3.3.3. The map t P(t) is continuous, concave, monotone increasing on R. 7→ Proof. Since µt is a probability measure on Σ and Ξr is a partition we get

log ∑ı Ξ µt([ı]) 0 = ∈ r for every r > 0 log r and by (3.3.7), Theorem 3.3.1(5) and (3.2.1)

t log ∑ı Ξ Aı P(t) = lim ∈ r k k . (3.3.11) r 0+ log r → Using this form it can be easily seen that t P(t) is continuous, concave and mono- 7→ tone increasing.

By Lemma 3.3.3, the potential ϕt depends continuously on t. Moreover, by (3.3.5), ϕ (i) ϕ (j) Ctτi j. Thus, the Perron-Frobenius operator | t − t | ≤ ∧ N 1 − ϕt(ii) (Tt(g))(i) = ∑ e g(ii) i=0 depends continuously on t. Hence, both the unique eigenfunction ht of Tt and the eigenmeasure νt of the dual operator Tt∗ depend continuously on t. Since dµt = htdνt, see [Bow08, Theorem 1.16], we got that t µ is continuous in weak*-topology. 7→ t Hence, by (3.3.9) and (3.3.10), t h and t χ are continuous on R. 7→ µt 7→ µt 3.3. Pressure for matrices with dominated splitting of index-1 71

Proposition 3.3.4. The map t P(t) is continuously differentiable on R. Moreover, for 7→ every t R ∈ dim µ = tP0(t) P(t), H t − and log Ai1 Ain lim k ··· k = P0(t) for µt-almost every i Σ. n ∞ log λ λ ∈ → i1 ··· in Proof. We recall [Heu98, Theorem 3.1]. That is, since µt is a Gibbs measure

q log ∑ı Ξr µt([ı]) τµt (q) = lim ∈ r 0+ log r → is differentiable at q = 1 and τµ0 t (1) = dimH µt. On the other hand, by (3.3.7) and (3.3.11) τ (q) = P(tq) P(t)q. µt − Hence, by taking the derivative at q = 1 we get that P(t) is differentiable for every t R 0 and ∈ \ { } dim µ = tP0(t) P(t). H t − Let us observe that by (3.3.6), (3.3.8) and (3.3.9)

log Ai1 E(σi) dµt(i) dim µt = t − k | k P(t). H log λ dµ (i) − R − i1 t Thus, R

log Ai1 E(σi) dµt(i) P0(t) = − k | k for every t = 0. log λ dµ (i) 6 R − i1 t Since t µt is continuous in weak*-topologyR we get that t P0(t) is continuous on 7→ 7→ R 0 . On the other hand, the left and right hand side limits of P (t) at t = 0 exist \ { } 0 and are equal. Thus, t P(t) is continuously differentiable on R. 7→ By Theorem 3.3.1(5), equation (3.3.4) and ergodicity of µt we get the last assertion of the proposition.

Let us observe that by the definition of pressure function (3.2.1), P(0) = 1 and − thus, µ0 corresponds to the Bernoulli measure on Σ with probabilities (λ0,..., λN 1). − That is, µ ([i ,..., i ]) = λ λ . 0 1 n i1 ··· in Lemma 3.3.5. For every finite set of matrices with dominated splitting of index-1, P (0) A 0 ≥ 1/d ,P (d ) 1/d . Moreover, P (0) > 1/d if and only if P (d ) < 1/d if and only if 0 0 0 ≤ 0 0 0 0 0 0 µ = µ . d0 6 0 Proof. By the definition of P(t),(3.2.1), P(d0) = 0, where d0 is defined in (3.2.2). To- gether with P(0) = 1 and the concavity and differentiability of P(t) (by Lemma 3.3.3 − and Proposition 3.3.4), we get P (0) 1/d , P (d ) 1/d . Moreover, P (0) > 1/d 0 ≥ 0 0 0 ≤ 0 0 0 if and only if P0(d0) < 1/d0. On the other hand, by Proposition 3.3.4

d log A 0 hµ i n d0 dimH µd = d0P0(d0) = lim k | k = for µd -a.e. i, 0 n ∞ 0 log λi χµd → |n 0 where in the last equation we used the deﬁnition of µd0 , the entropy and the Lyapunov exponent. Since dim µ = 1, if P (d ) < 1/d then µ = µ . Otherwise, by [Bow08, H 0 0 0 0 0 6 d0 72 Chapter 3. Pointwise regularity of parameterized afﬁne zipper fractal curves

Theorem 1.22], for every σ-invariant, ergodic measure ν on Σ,

hν hν 1 and = 1 if and only if ν = µ0. log λ dν(i) ≤ log λ dν(i) − i0 − i0

R hµ R Therefore, if P (d ) = 1/d then d0 = 1 and so µ = µ . 0 0 0 χµ d0 0 d0 Lemma 3.3.6. For every α [α , α ] ∈ min max

log Ai m dimH i Σ : lim inf k | k α inf tα P(t) (3.3.12) ∈ m ∞ log λi ≤ ≤ t 0 { − } n → |m o ≥ and

log Ai m dimH i Σ : lim sup k | k α inf tα P(t) (3.3.13) ∈ m ∞ log λi ≥ ≤ t 0 { − } n → |m o ≤ Proof. For simplicity, we use the notations

log Ai m log Ai m G = i Σ : lim inf k | k α and Gα = i Σ : lim sup k | k α . α m ∞ ( ∈ log λi m ≤ ) ( ∈ m ∞ log λi m ≥ ) → | → | Let ε > 0 be arbitrary but ﬁxed and let us deﬁne the following sets of cylinders:

log Aı Dr(ε) = [ı] Ξρ : 0 < ρ r and k k α + ε ∈ ≤ log λı ≤ n o and log Aı Dr(ε) = [ı] Ξρ : 0 < ρ r and k k α ε . ∈ ≤ log λı ≥ − n o By deﬁnition, Dr(ε) is a cover of Gα and respectively, Dr(ε) is a cover of Gα. Now let Cr(ε) and Cr(ε) be a disjoint set of cylinders such that

[ı] = [ı] and [ı] = [ı].

[ı] Dr(ε) [ı] Cr(ε) [ı] Dr(ε) [ı] C (ε) ∈[ ∈[ ∈[ ∈[r Then by (3.3.7) and the deﬁnition of C (ε), for any t 0 r ≥ αt P(t)+(1+t)ε (αt P(t)+(1+t)ε) r − (Gα) ∑ λı − H ≤ [ı] C (ε) ∈ r 1 ε t P(t) λmin− r ∑ Aı λı− ≤ [ı] C (ε) k k ∈ r 1 ε 1 ε Cλmin− r ∑ µt([ı]) Cλmin− r . ≤ [ı] C (ε) ≤ ∈ r Hence, αt P(t)+(1+t)ε(G ) = 0 for any t > 0 and any ε > 0, so (3.3.12) follows. The H − α proof of (3.3.13) is similar by using the cover Cr(ε) of Gα. We note that by the concavity of P

inf tα P(t) = inf tα P(t) , t R { − } t 0 { − } ∈ ≤ 3.4. Pointwise Hölder exponent for non-degenerate curves 73 for every and α [P (0), α ], ∈ 0 max inf tα P(t) = inf tα P(t) , t R { − } t 0 { − } ∈ ≥ for every α [α , P (0)]. ∈ min 0

3.4 Pointwise Hölder exponent for non-degenerate curves

First, let us define the natural projections π and Π from the symbolic space Σ to the unit interval [0, 1] and the curve Γ. We recall that we assumed that all the signatures of the affine zipper Definition 3.1.1 is 0, and all the matrices are invertible. Therefore,

∞ π(i) = lim gi gi (0) = λi γi (3.4.1) n ∞ 1 n ∑ n 1 n → ◦ · · · ◦ n=1 | − ∞ Π(i) = lim fi fi (0) = Ai ti . (3.4.2) n ∞ 1 n ∑ n 1 n → ◦ · · · ◦ n=1 | − Observe that by the definition of the linear parametrization v of Γ, v(π(i)) = Π(i). In the analysis of the pointwise Hölder exponent α, defined in (3.1.3), the points play important role which are far away symbolically but close on the self-affine curve. To be able to handle such points we introduce the following notation

i j+1 i j+1 min σ ∧ i N − 1, σ ∧ j 0 , if ii j+1 + 1 = ji j+1, { i j+1 ∧ i j+1 ∧ } ∧ ∧ i j = min σ ∧ i 0, σ ∧ j N − 1 , if ji j+1 + 1 = ii j+1, ∨  { ∧ ∧ } ∧ ∧  0, otherwise, where 0 denotes the (0, 0, . . . ) and N − 1 denotes the (N 1, N 1, . . . ) sequence. It − − is easy to see that there exists a constant K > 0 such that

1 K− (λi i j+i j + λj i j+i j ) π(i) π(j) K(λi i j+i j + λj i j+i j ). (3.4.3) | ∧ ∨ | ∧ ∨ ≤ | − | ≤ | ∧ ∨ | ∧ ∨ Hence, the distance on [0, 1] is not comparable with the distance on the symbolic space. More precisely, let T be the set of points on the symbolic space, which has tail 0 or N 1, i.e. i T if and only if there exists a k 0 such that σki = 0 or − ∈ ≥ σki = N − 1. So if π(σki) is too close to the set π(T) inﬁnitely often then we lose the symbolic control over the distance π(i) π(i ) , where i is such that π(i ) π(i) | − n | n n → as n ∞. → On the other hand, the symbolic control of the set Π(i) Π(in) is also far non- i j k i−j k trivial. In general, Π(i) Π(j) = Ai i j (Π(σ ∧ i) Π(σ ∧ j)) is not comparable i jk − i j k k | ∧ i j − i j k to Ai i j Π(σ ∧ i) Π(σ ∧ j) , unless Π(σ ∧ i) Π(σ ∧ j) M, where M is the k | ∧ k · k − k h − i ∈ multicone satisfying the Deﬁnition 3.1.2. Thus, in order to handle

log Π(i) Π(i ) lim inf k − n k n ∞ log π(i) π(i ) → | − n | we need that i is sufﬁciently far from the tail set T and also that the points Π(in) on Γ can be chosen such that Π(σi ji) Π(σi in i ) M. So we introduce a kind of h ∧ − ∧ n i ∈ 74 Chapter 3. Pointwise regularity of parameterized afﬁne zipper fractal curves

FIGURE 3.2: Local neighbourhood of points in Bn,l,m.

exceptional set B, where both of these requirements fail. We define B Σ such that ⊆ B = i Σ : n 1, l 1, m 1, K 0, k K { ∈ ∀ ≥ ∀ ≥ ∀ ≥ ∃ ≥ ∀ ≥ k k M(Π(σ i)) B (Π(σ i)) Γ (Γ k Γ k m Γ k m ) = ∅ , 1/n σ i l σ i l 1(ik+l 1)(N 1) σ i l 1(ik+l +1)0 \ ∩ \ | ∪ | − − − ∪ | − (3.4.4)o where Γ = f (Γ) for any finite length word ı Σ and M(Π(i)) is the cone centered ı ı ∈ ∗ at Π(i). We note that if i = 0 (or i = N 1) then we define Γ k m = ∅ l l σ i l 1(il 1)(N 1) − | − − − (or Γ k m = ∅ respectively). σ i l 1(il +1)0 In particular,| − B contains those points i, for which locally the curve Γ will leave the cone M very rapidly. In other words, let

B = i Σ : n,l,m { ∈ m m (M(Π(i)) B1/n(Π(i))) Γ (Γi l Γi l 1(il 1)(N 1) Γi l 1(il +1)0 ) = ∅ . \ ∩ \ | ∪ | − − − ∪ | − o and ∞ ∞ ∞ ∞ ∞ k Bn,m,l,K = σ− Bn,l,m and B = Bn,l,m,K. k\=K n\=1 l\=0 m\=0 K[=0 For a visualisation of the local neighbourhood of a point in Bn,l,m, see Figure 3.2. In particular, we are able to handle the pointwise Hölder exponents at π(i) outside of the set B and we show that B is small in some sense.

Lemma 3.4.1. Let us assume that is non-degenerate. Then there exist n 1, l 1, m 1 F ≥ ≥ ≥ and  finite length word with  = l, such that | | B [] = ∅. n,l,m ∩ Proof. Our first claim is that there exists a finite sequence ı such that A (z z ) h ı N − 0 i ∈ M. Suppose that this is not the case. That is, for every finite length word A (z h ı N − 3.4. Pointwise Hölder exponent for non-degenerate curves 75

c 1 c z0) M . Equivalently, for every ﬁnite length word ı, zN z0 Aı− (M ). Thus, i ∈ ∞ 1 c h − i ∈ zN z0 k=0 ı =k Aı− (M ), which contradicts to our non-degeneracy assump- h − i ∈ | | tion. T T Let us ﬁx an ı such that A (z z ) M. Then f (z ) M( f (z )). By continu- h ı N − 0 i ∈ ı N ∈ ı 0 ity, one can choose k 1 large enough such that for every i [ı0k], ≥ ∈ f (z ) M(Π(i)) ı N ∈ and

ı +k k 1 f (z ) Π(i) = A k (z Π(σ| | i)) A A diam(Γ) A (z z ) , k ı 0 − k k ı0 0 − k ≤ k ıkk 0k ≤ 2k ı N − 0 k where we used the fact that f0(z0) = z0. Then 1 Π(i) f (z ) A (z z ) f (z ) Π(i) > A (z z ) . k − ı N k ≥ k ı N − 0 k − k ı 0 − k 2k ı N − 0 k We get that for every i [ı0k] ∈

f (z ) M(Π(i)) B 1 (Π(i)) Γ (Γ k Γ k 1 Γ ) = ∅. ı N Aı(zN z0) ı0 ı0 − 10 ı ı 1(ı ı 1)N ∈ \ 2 k − k ∩ \ ∪ ∪ || |− | |− 6 k 2 By ﬁxing  := ı0 , l :=  , m := 1 and n := A (z z ) , we see that Bn,l,m [] = | | k ı N − 0 k ∩ ∅. l m Proposition 3.4.2. Let us assume that is non-degenerate. Then dim π(B) < 1. More- F P over, for any ν fully supported ergodic measure on Σ, ν(B) = 0.

Proof. By definition, B B , B B and B B . More- n,l,m ⊇ n+1,l,m n,l,m ⊇ n,l+1,m n,l,m ⊇ n,l,m+1 over, B = σ K B . In particular, σ 1B = B B . Thus, for n,l,m,K − n,l,m,0 − n,l,m,0 n,l,m,1 ⊇ n,l,m,0 every n 1 ≥ B $ (B ), (3.4.5) n,l,m,0 ⊆ ı n,l,m,0 ı =q | [| where $ (i) = ıi. Let n 1, l 1, m 1 be natural numbers and  be the finite ı 0 ≥ 0 ≥ 0 ≥ length word with  = l as in Lemma 3.4.1, then | | 0 ∞ k B [] = σ− B [] B [] = ∅. n0,l0,m0,0 ∩ n0,m0,l0 ∩ ⊆ n0,m0,l0 ∩ k\=0 Thus, B $ (B ). (3.4.6) n0,l0,m0,0 ⊆ ı n0,l0,m0,0 ı =l | | 0 ı[= 6 Hence, σpi / [] for every i B and for every p 1. Indeed, if there exists ∈ ∈ n0,l0,m0,0 ≥ i B and p 1 such that σpi [] then there exist a finite length word ı with ∈ n0,l0,m0,0 ≥ ∈ ı = p such that B [ı] = ∅. But by equations (3.4.5) and (3.4.6), | | ∩ 6 B $ (B ) $ ($ (B )) [ı ı ] n0,l0,m0,0 ⊆ ı1 n0,l0,m0,0 ⊆ ı1 ı2 n0,l0,m0,0 ⊆ 1 2 ı =p ı =p ı =l ı =p ı =l 1 1 | 2| 0 1 | 2| 0 | [| | [| ı[= | [| ı[= 26 26 76 Chapter 3. Pointwise regularity of parameterized affine zipper fractal curves which is a contradiction. But for any fully supported ergodic measure ν, ν([]) > 0 and therefore ν(Bn0,l0,m0,0) = 0. The second statement of the lemma follows by

∞ ∞ ∞ ( ) ( ) ( ) = ( ) = ν B inf ν Bn,l,m,K ∑ ν Bn0,l0,m0,K ∑ ν Bn0,l0,m0,0 0. ≤ n,l,m ≤ K= K= K[=0 0 0 To prove the ﬁrst assertion of the proposition, observe that by equation (3.4.6)

π(B ) g (π(B )). n0,l0,m0,0 ⊆ ı n0,l0,m0,0 ı =l | | 0 ı[= 6

Therefore, π(Bn ,l ,m ,0) is contained in the attractor Λ of the IFS gı ı =l , for which 0 0 0 | | 0 { } ı= 6 dimB Λ < 1. Hence,

dimP π(B) inf dimBπ(Bn,l,m,0) dimBπ(Bn ,l ,m ,0) dimB Λ < 1. ≤ n,l,m ≤ 0 0 0 ≤

Lemma 3.4.3. Let us assume that is non-degenerate. Then for every i Σ B F ∈ \

log Ai α(π(i)) lim sup k |n k. ≤ n +∞ log λi → |n ∞ Proof. Let i Σ B. Then there exist n 1, l 1, m 1 and a sequence k ∈ \ ≥ ≥ ≥ p p=1 such that k ∞ as p ∞ and p → →

M(Π(σkp i)) B (Π(σkp i)) \ 1/n ∩ Γ (Γ kp Γ kp m Γ kp m ) = ∅ (3.4.7) σ i l σ i l 1(ikp+l 1)(N 1) σ i l 1(ikp+l +1)0 \ | ∪ | − − − ∪ | − 6 Hence, there exists a sequence j such that k i j k + l, i j m, p p ≤ ∧ p ≤ p ∨ p ≤ 1 Π(σkp j ) M(Π(σkp i)) and Π(σkp j ) Π(σkp i) > . (3.4.8) p ∈ k p − k n Thus,

log Π(i) Π(j) log Π(i) Π(jp) α(π(i)) = lim inf k − k lim inf k − k π(j) π(i) log π(i) π(j) ≤ p +∞ log π(i) π(jp) → | − | → | − | kp kp log Ai k (Π(σ i) Π(σ jp)) = lim inf k | p − k , p +∞ log λ (π(σi jp+i jp i) π(σi jp+i jp j )) → i i jp+i jp ∧ ∨ ∧ ∨ p | | ∧ ∨ − | and by (3.3.3), (3.4.8),

m+l where d0 = (maxi λi) . Lemma 3.4.4. Let us assume that is non-degenerate. Then for every ergodic, σ-invariant, F ∞ k k fully supported measure µ on Σ such that ∑k=0(µ[0 ] + µ([N ]) is ﬁnite, then

log Ai α(π(i)) = lim k |n k for µ-a.e. i Σ. n +∞ log λi ∈ → |n Proof. By Proposition 3.4.2, we have that µ(B) = 0. Thus, by Lemma 3.4.3, for µ-a.e. i

log Ai α(π(i)) lim k |n k. ≤ n +∞ log λi → |n On the other hand, for every i Σ, ∈

log Π(i) Π(j) log Ai i j α(π(i)) = lim inf k − k lim inf k | ∧ k . j i π(j) π(i) log π(i) π(j) ≥ log λi i j+i j + log mini λi → | − | → | ∧ ∨ Hence, to verify the statement of the lemma, it is enough to show that

log λi i j lim | ∧ = 1 for µ-a.e. i. j i log λi i j+i j → | ∧ ∨ i j It is easy to see that from limj i i∨j = 0 follows the previous equation. Let → ∧

i jk 1 Rn = i : jk s. t. jk i as k ∞ and lim ∨ > ∃ → → k ∞ i j n → ∧ k In other words,

k/n k/n ∞ ∞ N b c b c Rn = [i1,..., ik, 0, . . . , 0] [i1,..., ik, N,..., N] K=0 k=K i ,...,i =0 ∪ \ [ 1 [k z }| { z }| { Therefore, for any µ ergodic σ-invariant measure and for every K 0 ≥ ∞ k/n k/n µ(Rn) ∑ (µ([0b c]) + µ([Nb c])). ≤ k=K

Since by assumption the sum on the right hand side is summable, we get µ(Rn) = 0 for every n 1. ≥ Lemma 3.4.5. Let us assume that is non-degenerate and satisﬁes (3.2.5). Then for every F i Σ ∈ log Ai α(π(i)) lim inf k |n k. ≥ n +∞ log λi → |n

Proof. Let us observe that by the zipper property fi(Π(0)) = fi 1(Π(N − 1)) for ev- − ery 1 i N 1. Moreover, for any i, j with ii j+1 = ji j+1 + 1, ≤ ≤ − ∧ ∧ i j+1 i j+1 i j = min σ ∧ i N − 1, σ ∧ j 0 . ∨ { ∧ ∧ } 78 Chapter 3. Pointwise regularity of parameterized afﬁne zipper fractal curves

Thus, if ii j+1 = ji j+1 + 1 ∧ ∧

Π(i) Π(j) = Π(i) Π(i i jii j+10) + Π(j i j ji j+1N − 1) Π(j) k − k k − | ∧ ∧ | ∧ ∧ − k i j+i j − i j+i j = Ai i j+i j (Π(σ ∧ ∨ i) Π(0)) + Ai i j+i j (Π(N 1) Π(σ ∧ ∨ j)) k | ∧ ∨ − | ∧ ∨ − k

( Ai i j+i j + Aj i j+i j )diam(Γ). (3.4.9) ≤ k | ∧ ∨ k k | ∧ ∨ k

The case ii j+1 = ji j+1 1 is similar, and if ii j+1 ji j+1 = 1 then i j = 0, so ∧ ∧ − | ∧ − ∧ | 6 ∨ (3.4.9) holds trivially. Moreover by (3.4.3), there exist constants K1, K2 > 0 such that for every i, j Σ ∈

log Π(i) Π(j) log K1 + log( Ai i j+i j + Aj i j+i j ) k − k − k | ∧ ∨ k k | ∧ ∨ k . (3.4.10) log π(i) π(j) ≥ log(λi i j+i j + λj i j+i j ) + log K2 | − | | ∧ ∨ | ∧ ∨ Therefore,

By Theorem 3.3.1(5) and (3.3.4), there exist C0 > 0 such that

i j A A E(i ) = A E(σ i i ) A i j E(i ) i i j+i j i i j+i j 0 i i j ∧ i j 0 σ ∧ i i j 0 k | ∧ ∨ k ≥ k | ∧ ∨ | k k | ∧ | | ∨ kk | ∨ | k C A A i j 0 i i j σ ∧ j i j ≥ k | ∧ kk | ∨ k and A A A i j j i j+i j j i j σ ∧ j i j k | ∧ ∨ k ≤ k | ∧ kk | ∨ k clearly. The other bounds are similar. But if ii j+1 = ji j+1 + 1 then Aσi jj = ∧ ∧ ∧ i j i j i j k | ∨ k A ∨ and A i j = A ∨ . Thus, by (3.2.5), 0 σ ∧ i i j N 1 k k k | ∨ k k − k log A i i j+i j log Ai n α(π(i)) lim inf k | ∧ ∨ k lim inf k | k. j i n +∞ ≥ log λi i j+i j ≥ log λi n → | ∧ ∨ → |

Proof of Theorem 3.2.1. First, we show that for -a.e. x, the local Hölder exponent is a N L constant. Since µ0 = λ1,..., λN , it is easy to see that π µ0 = [0,1]. Thus, it is { } ∗ L| 3.4. Pointwise Hölder exponent for non-degenerate curves 79 enough to show that for µ -a.e. i Σ, α(π(i)) is a constant. 0 ∈ But by Proposition 3.3.4, there exists α such that for µ0-a.e. i

log Ai α = limb k |n k. n +∞ log λi → |n By deﬁnition of Bernoulli measure,b ∞ µ ([0k]) + µ ([Nk]) = 1 + 1 . Thus, by ∑k=0 0 0 1 λ1 1 λN Lemma 3.4.4, α(π(i)) = α for µ -a.e. i, and by Lemma 3.3.5, we have− α − 1/d . 0 ≥ 0 We show now the lower bound for (3.2.4). By Lemma 3.3.3 and Proposition 3.3.4, the map t P (t) is continuous and non-increasing on R. Hence, for every β 7→ 0 b b ∈ (α , α ) there exists a t R such that P (t ) = β. By Proposition 3.3.4, there min max 0 ∈ 0 0 exists a µt0 Gibbs measure on Σ such that

log Ai n lim k | k = β for µt0 -a.e. i Σ. n +∞ log λi ∈ → |n It is easy to see that for any i and n 1, µ ([i ]) > 0. Thus, by Lemma 3.4.4, ≥ t0 |n α(π(i)) = β for µ -a.e. i Σ. t0 ∈ Observe that π : Σ [0, 1] is a ﬁnite to one Lipschitz-map. Thus, by [FH09, The- 7→ orem 2.8, Corollary 4.16], dim µ = dim µ π 1 for every t R. Therefore, by H t H t ◦ − ∈ Proposition 3.3.4

1 dim x [0, 1] : α(x) = β dim µ π− = t P0(t ) P(t ) = H { ∈ } ≥ H t0 ◦ 0 0 − 0 t0β P(t0) inf tβ P(t) . − ≥ t R { − } ∈ On the other hand, by Lemma 3.4.3

dim x [0, 1] : α(x) = β H { ∈ } max dim π(B), dim i Σ B : α(π(i)) = β ≤ { H H { ∈ \ }}

log Ai n max dimH π(B), dimH i Σ : lim sup k | k β ≤ ( ( ∈ n +∞ log λi n ≥ )) → |

max dimH π(B), inf tβ P(t) , ≤ t 0 { − } ≤ where in the last inequality we used Lemma 3.3.6. By Proposition 3.3.4, the function t tP (t) P(t) is continuous and P(0) = 1. 7→ 0 − − By Proposition 3.4.2, dimH π(B) < 1, thus, there exists an open neighbourhood of t = 0 such that for every t ( ρ, ρ), tP (t) P(t) > dim π(B). In other words, ∈ − 0 − P there exists an ε > 0 such that P (t) (α ε, α + ε) for every t ( ρ, ρ). Hence, for 0 ∈ − ∈ − every β [α, α + ε] there exists a t0 0 such that P0(t0) = β and inft 0 tβ P(t) = ∈ ≤ ≤ { − } t P (t ) P(t ) > dim π(B) which completes the proof of (3.2.4). 0 0 0 − 0 H b b Finally,b ifb (3.2.5) holds then by Lemma 3.4.5 and Lemma 3.3.6

dim x [0, 1] : α(x) = β H { ∈ } ≤

log Ai n dimH i Σ : lim inf k | k β inf tβ P(t) , n +∞ t 0 ( ∈ log λi n ≤ ) ≤ { − } → | ≥ which completes the proof. 80 Chapter 3. Pointwise regularity of parameterized afﬁne zipper fractal curves

3.5 Zippers with Assumption A

Now, we turn to the case when our affine zipper satisfies the Assumption A. We will show that in fact in this case the exceptional set B, introduced in (3.4.4) is empty. That is, there are no points, in which local neighbourhood, the curve leaves the cone rapidly. First, let us introduce a natural ordering on Σ . For any ı,  Σ with ı  = m ∗ ∈ ∗ ∧ and ı > m,  > m, let | | | | ı <  i < j . ⇔ m+1 m+1 Moreover, let Z := z ,..., z the endpoints of the curves f (Γ) and let Z := 1 { 0 N} i n f (z ), ı = n, k = 0, . . . , N 1 . { ı k | | − } For simplicity, let us denote fı(z0) by zı. Observe that by the Zipper property f (z ) = z . ı N ı ı 1(i ı 1) || |− | |− Proposition 3.5.1. Let us assume that is non-degenerate and satisfies the Assumption A. F Then B = ∅, where the set B is defined in (3.4.4).

Proof. It is enough to show that for every i Σ ∈ C(Π(i)) Γ = Γ, (3.5.1) ∩ which is equivalent to show that for every i, j Σ, Π(i) Π(j) C. ∈ h − i ∈ Since z z C and C is invariant w.r.t all of the matrices then for every ı Σ , h 0 − Ni ∈ ∈ ∗ z z = f (z ) f (z ) = A (z z ) C. ı ı ı 1(i ı 1) ı 0 ı N ı 0 N h − || |− | |− i h − i h − i ∈ Observe by convexity of C, for any three vectors x, y, w Rd, if x y C and ∈ h − i ∈ y w C then x w C. Thus, by Assumption A and the convexity of the cone, h − i ∈ h − i ∈ for every n 1, and for every ı <  Σ with ı =  = n, z z C. ≥ ∈ | | | | h ı − i ∈ Thus, for every i = j Σ and for every n 1, fi (z0) fj (z0) = zi zj 6 ∈ ≥ h |n − |n i h |n − |n i ∈ C. Since C is closed, by taking n tends to inﬁnity, we get that Π(i) Π(j) C. h − i ∈ Lemma 3.5.2. Let us assume that is non-degenerate and satisﬁes the Assumption A. Then F for any µ fully supported, ergodic, σ-invariant measure on Σ

log Π(i) Π(j) log Ai lim sup k − k lim k |n k for µ-a.e. i. log π(i) π(j) n +∞ log λ π(j) π(i) ≤ → i n → | − | | Proof. Observe that

log Π(i) Π(j) lim sup k − k = π(j) π(i) log π(i) π(j) → | − | i j+i j i j+i j log Ai i j+i j Π(σ ∧ ∨ i) Π(0) + Aj i j+i j Π( f rm[o] ) Π(σ ∧ ∨ j) lim sup k | ∧ ∨ − | ∧ ∨ −− − k. π(j) π(i) i j+i j i j+i j λi i j λi i j (π(σ ∧ ∨ i) 0) + λj i j (1 π(σ ∧ ∨ j)) → | | ∧ | ∨ − | ∨ − | By (3.5.1), Π(σi j+i ji) Π(0) , Π(N − 1) Π(σi j+i jj) C, therefore by (3.3.3) h ∧ ∨ − i h − ∧ ∨ i ∈ i j A ∨ k 0 k log Ai i j+i j + log 1 + i j log Π(i) Π(j) k | ∧ ∨ k AN∨ 1 lim sup k − k lim sup k − k . π(j) π(i) log π(i) π(j) ≤ j i log λi i j → | − | → | ∧ 3.5. Zippers with Assumption A 81

It is easy to see that for any fully supported, ergodic, σ-invariant measure µ, for µ-a.e. i i j lim sup ∨ = 0. j i i j + i j → ∧ ∨ Hence, by the previous inequality, the statement follows similarly as in Lemma 3.4.4 .

Proof of Theorem 3.2.2. By Lemma 3.4.4 and Lemma 3.5.2, for every t R ∈

log Ai n αr(π(i)) = lim k | k for µt-a.e. i Σ. n +∞ log λi ∈ → |n Thus, similarly to the proof of Theorem 3.2.1

1 dim x [0, 1] : α (x) = β dim µ π− = t P0(t ) P(t ) = H { ∈ r } ≥ H t0 ◦ 0 0 − 0 t0β P(t0) inf tβ P(t) , − ≥ t R { − } ∈ where t0 is deﬁned such that P0(t0) = β. On the other hand,

dim x [0, 1] : α (x) = β dim x [0, 1] : α(x) = β . H { ∈ r } ≤ H { ∈ } By Proposition 3.5.1, B = ∅, and similarly to the proof of Theorem 3.2.1,

dimH x [0, 1] : α(x) = β inf tβ P(t) , { ∈ } ≤ t 0 { − } ≤ for every β [αˆ , α ]. If Γ satisﬁes (3.2.5) then by Lemma 3.4.5 and Lemma 3.3.6 ∈ max

dimH x [0, 1] : α(x) = β inf tβ P(t) , { ∈ } ≤ t 0 { − } ≥ which completes the proof.

Now, we turn to the equivalence of the existence of pointwise regular Hölder exponents and the Assumption A. Before that, we introduce another property and we show that in fact all of them are equivalent. Denote cv(a, b) open line segment in Rd connecting two points a, b. Moreover, let us denote the orthogonal projection to a subspace θ by projθ and for a subspace θ let θ⊥ be the orthogonal complement of θ. For a point x and a subspace θ, let θ(x) = y Rd : x y θ . { ∈ − ∈ } Deﬁnition 3.5.3. We say that Z is well ordered on l G(d, d 1) if for any ı < ı < ı n ∈ − 1 2 3

proj (zı ) cv proj (zı ), proj (zı ) . (3.5.2) l⊥ 2 ∈ l⊥ 1 l⊥ 3 We say that Zn is well ordered if there exists a δ > 0 such that Zn is well ordered for all l B (F(Σ)). ∈ δ Let us recall that F : Σ G(d, d 1) is the Hölder-continuous function defined in 7→ − Theorem 3.3.1. So Bδ(F(Σ)) is the δ > 0 neighbourhood of all the possible subspaces on which the growth rate of the matrices is at most the second singular value. For a visualisation of the well ordered property, see Figure 3.3. Roughly speaking, the well ordered property on l G(d, d 1) means that the curve is parallel to l . The next ∈ − ⊥ lemma indeed verifies that the curve cannot turn back along l⊥. 82 Chapter 3. Pointwise regularity of parameterized affine zipper fractal curves

FIGURE 3.3: Well ordered property of Z on l G(d, d 1). n ∈ −

Lemma 3.5.4. Z is well ordered if and only if δ > 0, x Rd, l B (F(Σ)) either n ∃ ∀ ∈ ∀ ∈ δ cv(z , z ) l(x) = ∅ or cv(z , z ) l(x) = ∅, for every ı < ı < ı . ı1 ı2 ∩ ı2 ı3 ∩ 1 2 3 Proof. Fix ı < ı < ı 0, . . . , N 1 n. Suppose that Z is well ordered but for 1 2 3 ∈ { − } n all δ > 0 there exists l B (F(Σ)) and x Rd such that cv(z , z ) l(x) = ∅ and ∈ δ ∈ ı1 ı2 ∩ 6 cv(z , z ) l(x) = ∅. Thus, ı2 ı3 ∩ 6

proj (x) cv proj (zı ), proj (zı ) cv proj (zı ), proj (zı ) l⊥ ∈ l⊥ 2 l⊥ 1 ∩ l⊥ 2 l⊥ 3 Since the right hand side is open, and non-empty line segment, therefore

proj (zı ) / cv proj (zı ), proj (zı ) , l⊥ 2 ∈ l⊥ 1 l⊥ 3 which is a contradiction. On the other hand, suppose that Zn satisfy the assumption of the lemma but not well ordered. Then for every δ > 0 there exists an l Bδ(F(Σ)) such that proj (zı ) / ∈ l⊥ 2 ∈ cv proj (zı ), proj (zı ) . Since Bδ(F(Σ)) is open, there exists an l0 Bδ(F(Σ)) for l⊥ 1 l⊥ 3 ∈ which dist(proj (zı ), cv proj (zı ), proj (zı ) ) > 0. l0⊥ 2 l0⊥ 1 l0⊥ 3 Thus, there exists x Rd that cv(z , z ) l (x) = ∅ and cv(z , z ) l (x) = ∅, ∈ ı1 ı2 ∩ 0 6 ı3 ı2 ∩ 0 6 which is again a contradiction.

The next lemma gives us a method to check the well ordered property. Lemma 3.5.5. Z is well ordered if and only if for every n 0 Z is well ordered. 0 ≥ n Proof. The if part is trivial. N 1 By deﬁnition, Zn = − fk(Zn 1). By Lemma 3.5.4, if Zn 1 is well ordered k=0 − − then there exists δ > 0 such that for every l B (F(Σ)) and for every x Rd ei- S ∈ δ ∈ ther cv(z , z ) l(x) = ∅ or cv(z , z ) l(x) = ∅. Thus, in particular for every ı1 ı2 ∩ ı2 ı3 ∩ 3.5. Zippers with Assumption A 83 l B (F([k])). By Theorem 3.3.1(2), there exists δ > 0 such that A B (F([k])) ∈ δ 0 k δ ⊇ B (F(Σ)). Thus, for every l B (F(Σ)) and for every x Rd, δ0 ∈ δ0 ∈ either cv( f (z ), f (z )) l(x) = ∅ or cv( f (z ), f (z )) l(x) = ∅ (3.5.3) k ı1 k ı2 ∩ k ı2 k ı3 ∩ for every zı , zı , zı Zn 1 with ı1 < ı2 < ı3. 1 2 3 ∈ − Let us suppose that Zn is not well ordered for some n. Hence, there exists a minimal n such that Zn 1 is well ordered but Zn is not. By Lemma 3.5.4, for every − δ > δ > 0 there exist  <  <  0, . . . , N 1 n+1, l B (F(Σ)) and x Rd 0 1 2 3 ∈ { − } 0 ∈ δ ∈

cv(z , z ) l0(x) = ∅ and cv(z , z ) l0(x) = ∅. 1 2 ∩ 6 2 3 ∩ 6

Since (3.5.3) holds for every k = 0, . . . , N 1, there are k < m such that z fk(Zn 1) − 1 ∈ − and z fm(Zn 1). On the other hand by (3.5.3), one of the endpoints of fk(Zn 1) 3 ∈ − − (and fm(Zn 1)) must be on the same side of l0(x), where z (and z respectively) is. − 1 3 Denote these endpoints by z and z . Observe that z = z . Indeed, if z = z a0 b0 a0 6 b0 a0 b0 then k = m 1, and thus either z fk(Zn 1) or z fm(Zn 1). Hence, but z is − 2 ∈ − 2 ∈ − 2 z z z z l (x) separated from 1 , 3 , a0 , b0 by the plane 0 , which cannot happen by (3.5.3). Moreover,by (3.5.3), one of the endpoints of f( ) (Zn 1) is on the same side of 2 0 − l0(x) with z , denote it by zc . But the endpoints of fp(Zn 1) are the elements of Z0, 2 0 − moreover, a0 < c0 < b0, which contradicts to the well ordered property of Z0. Theorem 3.5.6. Let be a non-degenerate system. Then the following three statements are F equivalent

1. S satisﬁes Assumption A,

2. for -a.e. x, α (x) exists, L r 3.Z 0 satisﬁes the well-ordered property. Proof of Theorem 3.5.6(1) Theorem 3.5.6(2). Similarly, to the begining of the proof of ⇒ N Theorem 3.2.1, π µ0 = [0,1], where µ0 = λ1,..., λN , which is fully supported, ∗ L| { } σ-invariant, ergodic measure. Moreover, ∞ µ ([0k]) + µ ([Nk]) = 1 + 1 . ∑k=0 0 0 1 λ1 1 λN Thus, by Lemma 3.4.4 − −

log Π(i) Π(j) log Ai n lim inf k − k = lim k | k for µ0-a.e. i Σ. π(j) π(i) log π(i) π(j) n +∞ log λi ∈ → | − | → |n But since satisﬁes Assumption A, by using Lemma 3.5.2, F

log Π(i) Π(j) log Ai n lim sup k − k lim k | k for µ0-a.e. i Σ, log π(i) π(j) n +∞ log λ π(j) π(i) ≤ → i n ∈ → | − | | which completes the proof.

Proof of Theorem 3.5.6(2) Theorem 3.5.6(3). Let us argue by contradiction. Assume that ⇒ α (x) exists for -a.e. x but there exists Z , n 0, which does not satisfy the well- r L n ≥ ordered property. By Lemma 3.5.5, Z0 does not satisfy the well ordered property. By d Lemma 3.5.4, let l F(Σ), x R and zi 1, zi, zi+1 Z0 be such that cv(zi 1, zi) ∈ ∈ − ∈ − ∩ l(x) = ∅ and cv(z , z ) l(x) = ∅. By continuity of the curve Γ, there exist i, j Σ 6 i i+1 ∩ 6 ∈ such that i = i 1 = i = j , j = 0 and Π(i) Π(j) l(x ) with some x Rd. 1 − 6 1 2 6 − ∈ 0 0 ∈ Hence, Π(i) Π(j) l. By definition, there exists a k Σ such that F(k) = l. By h − i ⊂ ∈ 84 Chapter 3. Pointwise regularity of parameterized affine zipper fractal curves using the continuity of F : Σ G(d, d 1), Γ and Π : Σ Γ, one can choose n, m 7→ − 7→ sufficiently large, such that for every i [i ] and k [k ], 0 ∈ |n 0 ∈ |m

F(k0)(Π(i0)) Γ = ∅. (3.5.4) ∩ j1 j2 6 By ergodicity, for µ -a.e. i, σpi [k ,..., k , i ,..., i ] for inﬁnitely many p 0, 0 ∈ m 1 1 n ≥ where k = k ,..., k . Let us denote this subsequence by p . Let k be the sequence |m 1 m k k such that k [k ,..., k , i ,..., i ]. k ∈ 1 m pk 1 By (3.5.4), there exists a sequence j such that j i = p + m, σpk+mj [j j ], { k} k ∧ k k ∈ 1 2 and Π(σpk+mj ) Π(σpk+mi) F(k ). By construction, σpk+mj σpk+mi = 0 and k − ∈ k k ∧ σpk+mj σpk+mi = 0, and hence there exists a constant c > 0 such that k ∨ Π(σpk+mj ) Π(σpk+mi) > c. Therefore for µ -a.e. i k k − k 0

log Π(i) Π(j) log Π(i) Π(jk) αr(π(i)) = lim k − k = lim k − k π(j) π(i) log π(i) π(j) k +∞ log π(i) π(jk) → | − | → | − | pk+m pk+m log Ai p +m (Π(σ i) Π(σ jk)) = lim k | k − k pk+m pk+m k +∞ log λi + (π(σ i) π(σ jk)) → | |pk m − |

log Ai p +m F(kk) log α2(Ai p +m ) lim k | k | k lim | k ≥ k +∞ log λi + ≥ k +∞ log λi + → |pk m → |pk m

log τ log Ai p +m log τ + lim k | k k = + α(π(i)), ≥ χµ0 k +∞ log λi + χµ0 − → |pk m − (3.3.10) where Theorem 3.3.1(4), Theorem 3.3.1(6) and Lemma 3.4.4. But log τ/χ > − µ0 0, which is a contradiction.

Let us recall that for any 0 = x Rd, v denotes the unique 1-dimensional d 1 6 ∈ h i subspace in PR − such that v v . Also, any V G(d, d 1) can be identified with ∈ h i d∈1 − d 1 a d 2 dimensional, closed submanifold V˜ of PR − such that V˜ = θ PR − : θ − { ∈ d ⊂1 V . Also, for a subset B G(d, d 1) we can identify it with a subset B˜ of PR − } d 1⊂ − such that B˜ = θ PR − : θ V B . { ∈ ⊂ ∈ } Proof of Theorem 3.5.6(3) Theorem 3.5.6(1). Suppose that Z satisfies the well ordered ⇒ 0 property. By Lemma 3.5.5, Z satisfies the well-ordered property for every n 0 n ≥ and thus, we may assume that z z / F](Σ) for every z , z Z . Indeed, if h ı − i ∈ ı  ∈ n zı z F(i) for some i Σ then one could find z , z , z Zn+1 such that h − i ∈ ∈ ı10 ı20 ı30 ∈ ı10 < ı20 < ı30 , zı0 = zı, zı0 = z and g1 3 proj (z ) / cv(proj (z ), proj (z )). F(i)⊥ ı20 ∈ F(i)⊥ ı10 F(i)⊥ ı30

d 1 So, for every ı,  Σ there exists open, connected component C of PR − F](Σ) ∈ ∗ ı, \ such that z z C . h ı − i ∈ ı, Then for any ı < ı < ı , C = C = C . Indeed, if C = C then there 1 2 3 ı1,ı2 ı1,ı3 ı2,ı3 ı1,ı2 6 ı1,ı3 exists F(i), which separates zı zı and zı zı . But then, for F(i)⊥, proj (zı ) / h 2 − 1 i h 3 − 1 i F(i)⊥ 2 ∈ cv(proj (zı ), proj (zı )), which cannot happen by deﬁnition of well ordered F(i)⊥ 1 F(i)⊥ 3 property.g Therefore, there exists a unique open,connected component C such that z z h ı − i ∈ C for every ı,  Σ . But, for any i Σ, since z z / F](Σ) ∈ ∗ ∈ h N − 0i ∈

lim Ai (zN z0) = E(i), n +∞ |n → h − i 3.6. An example, de Rham’s curve 85 hence, E(Σ) C. Thus, for any multicone M, for which the dominated splitting ⊂ condition of index-1 holds, the cone M C is invariant, i.e. A (M C) Mo C. ∩ i ∩ ⊂ ∩ On the other hand, by z z C, one can extend M C such that z z h N − 0i ∈ ∩ h N − 0i ∈ M C and M C remains invariant. ∩ ∩

3.6 An example, de Rham’s curve

Now we show an application for our main theorems. Recall, the well-known de Rham’s curve [Rha47; Rha56; Rha59] in R2 is the attractor of the affine zipper defined by the functions f and f given in (1.2.2), where ω (0, 1/2) is the parameter of the curve. 0 1 ∈ In the introductory Subsection 1.2.2 a linear parametrization v : [0, 1] R2 was also 7→ given, see (1.2.3) and a visualization of the curve, see Figure 1.3. Protasov [Pro04; Pro06] proved in a more general context that the set of points x ∈ [0, 1] for which α(x) = β has full measure only if β = α, otherwise it has zero measure. Just recently, Okamura [Oka16] bounds α(x) for Lebesgue typical points allowing in the definition (1.2.3) more than two functions and alsob non-linear functions under some conditions. We show that with a suitable coordinate transform the matrices A0 and A1 satisfy Assumption A and hence, our results are applicable.

Lemma 3.6.1. For every ω (0, 1/3) (1/3, 1/2) there exists a coordinate transform 1 ∈ ∪ D(ω) such that D(ω)− AiD(ω) has strictly positive entries for i = 0, 1. Proof. For ε > 0 and 0 < δ < 1 deﬁne the coordinate transform matrices

1 ε 1 δ Dε = and Dδ = − . ε 1 δ 1 − e b 1 1 Elementary calculations show that the matrices A0 = Dε− A0Dε and A1 = Dε− A1Dε have strictly positive entries whenever e e e e e e 1 1 < ω < . 3 ε 2 ε ε2 − − − The largest possible interval (1/3, 1/2) is attained when ε is arbitrarily small. Very 1 1 similar calculations show that the entries of A0 = Dδ− A0Dδ and A1 = Dδ− A1Dδ are strictly positive whenever δ b 1 b b b b b < ω < , 1 + 3δ 3 + δ which gives the open interval (0, 1/3). Also trivial calculations show that A = k ik1 Ai 1 = Ai 1, i = 0, 1. k k k k e Let us recallb that in this case P(t) has the form

1 t P(t) = lim − log Aı , n +∞ n ∑ → log 2 ı =n k k | | and P(t) P(t) αmin = lim and αmax = lim . t +∞ t t ∞ t → →− Proposition 3.6.2. For every ω (0, 1/4) (1/4, 1/3) (1/3, 1/2) the de Rham function ∈ ∪ ∪ v : [0, 1] R2, deﬁned in (1.2.3), the following are true 7→ 86 Chapter 3. Pointwise regularity of parameterized afﬁne zipper fractal curves

1. v is differentiable for Lebesgue-almost every x [0, 1] with derivative vector equal to ∈ zero,

2. Let be the set of [0, 1] such that v is not differentiable. Then dim = τ P(τ) > N H N − 0, where τ R is chosen such that P (τ) = 1, ∈ 0 1 1 N N 3. dimH Π µ0 < 1, where µ0 = , equidistributed measure on Σ = 0, 1 and ∗ 2 2 { } Π is the natural projection from Σ to v([0, 1]). 4. for every β [α , α ] ∈ min max dim x [0, 1] : α(x) = β = dim x [0, 1] : α (x) = β H { ∈ } H { ∈ r } = inf tβ P(t) . t R{ − } ∈ For ω = 1/4 the de Rham curve is a smooth curve, namely a parabola arc. For ω = 1/3, the matrices does not satisfy the dominated splitting condition. For this case, we refer to the work of Nikitin [Nik04].

Proof. By Lemma 3.6.1, we are able to apply Theorem 3.2.2 and Theorem 3.2.3 for ω = 1/3. It is easy to see that (1.2.2) satisﬁes (3.2.5). Thus, by Theorem 3.2.2, the 6 statement (4) of the proposition follows. On the other hand, let be the set, where v is not differentiable. Then N x [0, 1] : α(x) < 1 x [0, 1] : α(x) 1 . { ∈ } ⊆ N ⊆ { ∈ ≤ }

Thus, dimH = inft R t P(t) . N ∈ { − } Now, we prove that there exists τ R such that P0(τ) = 1. Observe that M = ∈ T A0 + A1 is a stochastic matrix with left and right eigenvectors p = (p1, p2) and T e = (1, 1) , respectively, corresponding to eigenvalue 1 and pi > 0, p1 + p2 = 1. There exists a constant c > 0 such that for every ı Σ ∈ ∗ 1 T T c− p A e A cp A e, ı ≤ k ık ≤ ı and therefore T T n ∑ p Aıe = p (A0 + A1) e. ı =n | | Thus P(1) = 0, and

T T µ1([i n]) = p Ai e = p Ai1 ... Ain e and | |n 1 µ ([i ]) = for every i Σ. 0 |n 2n ∈ Simple calculations show that,

1 1 (1 2ω)ω (1 2ω)2 µ ([00]) = , A A (1, 1)T = ω2 + − + − , 1 2 2 0 0 2 2 which is not equal to 1/4 if ω = 1/4 or ω = 1/2. Thus, for ω = 1/4 and ω = 1/2, 6 6 6 6 µ = µ , and by Lemma 3.3.5, P (1) < 1 < P (0). Since t P (t) is continuous, there 0 6 1 0 0 7→ 0 exists τ such that P (τ) = 1 and therefore dim = τ P(τ) > 0, which completes 0 H N − (2). 3.6. An example, de Rham’s curve 87

On the other hand, by Theorem 3.2.3, αr(x) = P0(0) > 1 for Lebesgue almost every x [0, 1] and therefore v is differentiable with derivative vector 0. This implies ∈ (1). Finally, we show statement (3) of the proposition. By using the classical result of Young

log Π µ0(Br(x)) dimH Π µ0 = lim inf ∗ for π µ0-a.e. x Γ = v([0, 1]). ∗ r 0+ log r ∗ ∈ → R For an i Σ and r let n 1 be such that Ai n r < Ai n 1 . Hence, ∈ ∈ ≥ k | k ≤ k | − k Π([i ]) B (Π(i)) and |n ⊆ r

Remark 3.6.3. Finally, we remark that in case of general signature vector, one may modify the deﬁnition of i j to ∨ i j+1 i j+1 min σ ∧ i N − 1, σ ∧ j 0 , ii j+1 + 1 = ji j+1 & εii j+1 = 0 & εji j+1 = 0, { i j+1 ∧ i j+1 ∧ } ∧ ∧ ∧ ∧ min σ ∧ i 0, σ ∧ j 0 , ii j+1 + 1 = ji j+1 & εii j+1 = 1 & εji j+1 = 0,  { ∧ ∧ } ∧ ∧ ∧ ∧ i j+1 i j+1 −  min σ ∧ i 0, σ ∧ j N 1 , ii j+1 + 1 = ji j+1 & εii j+1 = 1 & εji j+1 = 1,  ∧ ∧  { i j+1 ∧ i ∧j+1 } ∧ ∧  min σ ∧ i N − 1, σ ∧ j N − 1 , ii j+1 + 1 = ji j+1 & εii j+1 = 0 & εji j+1 = 1,  { ∧ ∧ } ∧ ∧ ∧ ∧ =  i j+1 N − i j+1 + = = = i j  min σ ∧ j 1, σ ∧ i 0 , ji j+1 1 ii j+1 & εji j+1 0 & εii j+1 0, ∨  { ∧ ∧ } ∧ ∧ ∧ ∧  i j+1 i j+1  min σ ∧ j 0, σ ∧ i 0 , ji j+1 + 1 = ii j+1 & εji j+1 = 1 & εii j+1 = 0, { i j+1 ∧ i j+1 ∧ } ∧ ∧ ∧ ∧ min σ ∧ j 0, σ ∧ i N − 1 , ji j+1 + 1 = ii j+1 & εji j+1 = 1 & εii j+1 = 1,  { ∧ ∧ } ∧ ∧ ∧ ∧  min σi j+1j N − 1, σi j+1i N − 1 , j + 1 = i & ε = 0 & ε = 1,  ∧ ∧ i j+1 i j+1 ji j+1 ii j+1  { ∧ ∧ } ∧ ∧ ∧ ∧  0, otherwise.    Open problems In this chapter we parametrized the points of zippers, but for zippers like in the example of Subsection 2.7.6 we can consider the attractor as the graph of a function f : [0, 1] [0, 1]. Now we can ask similar questions about f , but the methods re- → quired are different.

1. Perform multifractal analysis for the pointwise Hölder exponent of f .

2. Consider the level sets E := y : f (y) = x . From our dimension result, we x { } can determine the Hausdorff dimension of a level set for typical x. However, this is not the same value for all x [0, 1]. By keeping track of the codes of ∈ vertical columns, the goal is to determine the different values that dimH Ex can attain, and then study the map

α dim x [0, 1] : dim E = α . 7→ H{ ∈ H x }

Chapter 4

Distances in random and evolving Apollonian networks

This chapter is based on the article [KKV16] written jointly with Júlia Komjáthy and Lajos Vágó. The paper also contains results about the degree distribution and clustering coefﬁcient. However, I did not participate actively in this part, therefore it is left out of the thesis.

4.1 Deﬁnitions and notations

We now deﬁne the models and introduce necessary notation to state the results.

Random Apollonian networks

A random Apollonian network (RAN) in d dimensions, denoted RANd(n), is deﬁned as follows. The graph at step n = 0 consists of d + 2 vertices, embedded in Rd in such a way that d + 1 of them form a d-dimensional simplex, and the (d + 2)-th vertex is located in the interior of this simplex, connected to all of the vertices of the simplex. This vertex in the interior forms d + 1 d-simplices with the other vertices: initially we set the status of these d-simplices ‘active’, and call them active cliques. For n 1, pick ≥ an active clique C of RAN (n 1) uniformly at random (u.a.r.), insert a new vertex v n d − n in the interior of Cn and connect vn with all the vertices of Cn. The newly added vertex vn forms new cliques with each possible choice of d vertices of Cn. Set the status of the clique Cn ‘inactive’, and the status of the newly formed d-simplices ‘active’. The resulting graph is RANd(n). At each step n a RANd(n) has n + d + 2 vertices and nd + d + 1 active cliques.

Evolutionary Apollonian networks Given a sequence of occupation parameters q ∞ , 0 q 1, an evolving Apol- { n}n=1 ≤ n ≤ lonian network (EAN) EAN (n, q ) = EAN (n) in d dimensions can be constructed d { n} d iteratively as follows. The initial graph is the same as for a RANd(0) and there are d + 1 active cliques. For n 1, choose each active clique of EAN (n 1) indepen- ≥ d − dently of each other with probability q . The set of chosen cliques becomes inactive n Cn (the non-picked active cliques stay active) and for every clique C we place a new ∈ Cn vertex vn(C) in the interior of C that we connect to all vertices of C. This new vertex vn(C) together with all possible choices of d vertices from C forms d + 1 new cliques: these cliques are added to the set of active cliques for every C . The result- ∈ Cn ing graph is EANd(n). The set of inactive vertices after n steps is denoted V(n) and N(n) = V(n) . | | 90 Chapter 4. Distances in random and evolving Apollonian networks

The case q q was studied in [ZRZ06] where it was further suggested that for n ≡ q 0 the graph is similar to a RAN (n). We prove their conjecture in [KKV16] → d by showing that EANs obey the same power law exponent as RANs if qn 0 and ∞ → ∑n=0 qn = ∞. Remark 4.1.1. Note that both in the RAN and EAN models, there is a one-to-one correspondence between cliques and vertices/future vertices: vertex v corresponds to the clique C that became inactive when v was placed in the interior of the d-simplex corresponding to C. In this respect, we call vertices that are already present in the graph inactive vertices, and we refer to active cliques as active vertices: this notation means that these vertices are not yet present in the graph, but might become present in the next step of the dynamics.

Notation

Fix n and consider two ‘active’ or ‘inactive’ vertices u and v from RANd(n) or EANd(n). Recall that Hopd(n, u, v) is the hopcount between the vertices u and v, i.e., the number of edges on (one of) the shortest paths between u and v. The ﬂooding time Floodd(n, u) and diameter Diamd(n) are

Floodd(n, u) = max Hop (n, u, v) and Diamd(n) = max Hop (n, u, v). (4.1.1) v d u,v d

Whenever possible d, u and v are suppressed from the notation. We deﬁne Dv(n) as the degree of vertex v after the n-th step. d+1 Let (Xi)i=1 be a collection of independent geometrically distributed random vari- i ables with success probability d+1 for Xi. Deﬁne the sum

d+1 Yd := ∑ Xi. (4.1.2) i=1

Yd is commonly referred to as a full coupon collector block in a coupon collector problem with d + 1 coupons. Denote the expectation and variance of Yd by

2 2 µd := E[Yd] = (d + 1)H(d + 1), σd := D [Yd] , (4.1.3)

d where H(d) = ∑i=1 1/i. The Large Deviation rate function of Yd is given by

λYd Id(x) := sup λx log E e . (4.1.4) λ R − ∈ n h io

The rate function Id(x) has no explicit form. It can be computed numerically from

λ (x)Y I (x) = λ∗(x) x log E e ∗ d , d · − h i ∂ λYd where λ∗(x) is the unique solution to the equation ∂λ log E e = x and d i log E eλYd = log d! d log(d + 1) + (d + 1)λ log 1 eλ . − − ∑ − d + 1 h i i=1 The following is needed for the ﬂooding time and diameter. Consider the function

d + 1 d f (c) := c c log c . (4.1.5) d − d − d + 1 4.2. Main results 91

Notice that f (c) is the rate function of a Poi( d+1 ) random variable. Thus for c > − d d d+1 the equation f (c) = 1 has a unique solution which we denote by c . Finally we d d − d introduce c˜ µ g(α, β) := 1 + f (αc˜ ) αβ d I d .e (4.1.6) d d − µ d β d A sequence of events happens with high probability (w.h.p.) if lim P( ) = 1. En n En Note that ‘with high probability’ is the same as ‘asymptotically almost surely’. We further deﬁne for an event A and a σ-algebra the conditional probability P(A ) = F |F E[11 ], where 11 is the indicator of the event A, i.e., it takes value 1 if A holds and A|F A 0 otherwise. We will sometimes replace by a list of random variables, in this case F we drop the σ-algebra notation and list the random variables in the conditioning, and this means conditional on the σ-algebra generated by this list of random variables.

Structure of the chapter In Section 4.2 we state our main results and discuss their relation to other results in the ﬁeld. Section 4.3 contains the most important combinatorial observations about the structure of RANs: we work out an approach of coding the vertices of the graph that enables us to compare the structure of the RAN to a branching process and further, the distance between any two vertices in the graph is given entirely by the coding of these vertices. We also give a short sketch of proofs related to distances in this section. Then we prove rigorously the distance-related theorems in Section 4.4.

4.2 Main results

The ﬁrst theorem describes the asymptotic behavior of typical distances in RANd(n). Theorem 4.2.1 (Typical distances in RANs). The hopcount between two active vertices chosen u.a.r. in a RANd(n) satisﬁes a Central Limit Theorem (CLT) of the form

2 d+1 Hopd(n) µ d log n d − d Z, (4.2.1) σ2+µ −→ d d d+1 n 2 µ3 d log r d 2 where µd, σd as in (4.1.3) and Z is a standard normal random variable. Further, the same CLT is satisﬁed for the distance between two inactive vertices that are picked independently with the size-biased probabilities given by

(d 1)D (n) d2 + d + 2 P(v is chosen D (n)) = − v − , (4.2.2) | v dn + d + 1 where Dv(n) is the degree of the inactive vertex v.

Remark 4.2.2. Active vertices are not physically present in the graph. The distance is deﬁned between them so that we view them as inactive vertices with their initial d + 1 edges present in the graph.

The next theorem describes the asymptotic behaviour of the ﬂooding time and the diameter:

Theorem 4.2.3 (Diameter and flooding time in RANs). Let u denote either an active vertex chosen u.a.r. or an inactive vertex chosen according to the size-biased distribution 92 Chapter 4. Distances in random and evolving Apollonian networks given in (4.2.2). Then as n ∞ → Diam (n) P c˜ d 2α˜ β˜ d , log n −→ µ d (4.2.3) Flood (n, u) P 1 d + 1 d + α˜ β˜c˜ , log n −→ µ d d d where (α˜ , β˜) (0, 1] [1, µd ] is the optimal solution of the maximization problem with the ∈ × d+1 following constraint: max αβ : g(α, β) = 0 . (4.2.4) { } Remark 4.2.4. Observe that the set of (α, β) pairs that satisfy the constraint in (4.2.4) is non-empty since for α = β = 1 by definition f (c˜ ) = 1 and I (µ ) = 0. The fact that d − d d the pair (α˜ , β˜) is unique is proved in Lemma 4.4.9. The maximization problem is also equivalent to first defining the g(α(x), β(x)) := sup g(α, β) : αβ = x and then α,β{ } choosing the unique x with g(α(x), β(x)) = 0, where the existence and uniqueness of such x follows from the fact that g(α(x), β(x)) strictly monoton decreases in x and continuity. This is shown in Claim 4.4.8.

We conclude with the asymptotic behavior of the typical distances in EANd(n). Theorem 4.2.5 (Typical distances in EANs). Assume that the sequence of occupation parameters qn satisfies ∑n N qn = ∞ and ∑n N qn(1 qn) = ∞. Then, the hopcount { } ∈ ∈ − between two active vertices chosen u.a.r. in a EANd(n) satisfies a central limit theorem of the form n 2 Hopd(n) µ ∑ qi − d d i=1 Z, (4.2.5) σ2+µ n −→ d d q ( q ) 2 µ3 ∑ i 1 i s d i=1 − 2 where µd, σd as in (4.1.3) and Z is a standard normal random variable. Further, the same CLT is satisfied for the distance between two inactive vertices that are chosen independently with the size-biased probabilities given by

(d 1)D (n) d2 + d + 2 P (v is chosen D (n), N(n)) = − v − . (4.2.6) | v d(N(n) d 2) + d + 1 − − Remark 4.2.6. Note that in this theorem qn might or might not tend to 0. The second criterion rules out the case when q 1 so fast that the graph becomes essentially n → deterministic. Further, the statements of Theorems 4.2.1, 4.2.5 also stay valid if one of the vertices is an active vertex chosen uniformly at random and the other vertex is inactive chosen according to the distribution given in (4.2.2) and (4.2.6), respectively.

4.2.1 Related literature The statements of Theorem 4.2.1 are in agreement with previous results. In particular, in [ZRC06] the authors estimate the average path length, i.e., the hopcount averaged over all pairs of vertices, and they show that it scales logarithmically with the size of the network. A more reﬁned claim is obtained by Albenque and Marckert [AM08] concerning the hopcount in two dimensions. They prove that

Hop(n) P 1. 6/11 log n −→ 4.3. Structure of RANs and EANs 93

The constant 6/11 is the same as 2(d + 1)/(dµd) for d = 2. They use the previously mentioned notion of stack triangulations to derive the result from a CLT similar to the one in Theorem 4.2.1. We show an alternative approach using weaker results. The CLT for distances in RANs and EANs is novel. Central limit theorems of the form (4.2.1) for the hopcount have been proven with the addition of exponential or general edge weights for various other random graph models, known usually under the name first passage percolation. Janson [Jan99] anal- ysed distances in the complete graphs with independent and identically distributed (i.i.d.) exponential edge weights. In a series of papers Bhamidi, van der Hofstad and Hooghiemstra determine typical distances and prove CLT for the hopcount for the Erd˝os-Rényirandom graph [BHH11], the stochastic mean-field model [BHH12b], the configuration model with finite variance degrees [BHH10] and quite recently for the configuration model [BHH12a] with arbitrary i.i.d. edge weights from a continuous distribution. Inhomogeneous random graphs are handled by Bollobás, Janson and Ri- ordan [BJR07; KK15]. Note that in all these models the edges have random weights, while in RANs and EANs all edge weights are 1. The reason for this similarity is hid- den in the fact that all these models have an underlying branching process approximation, and the CLT valid for the branching process implies CLT for the hopcount on the graph. Further, there are some previous bounds known about the diameter of RANs: Frieze and Tsourakakis [FT12] establishes the upper bound 2c˜2 log n for RAN2(n). They use a result of Broutin and Devroye [BD06] that, combined with the branching process approximation of the structure of RANs we describe in this paper, actually implicitly gives the 2c˜d log n upper bound for all d. Just recently and independently from our work other methods were used to determine the diameter. In [Ebr+13] Ebrahimzadeh et al. apply the result of [BD06] in an elaborate way, while Cooper and Frieze in [CFU14] use a more analytical approach solving recurrence relations. We emphasize that the methods in [CFU14; Ebr+13] and in the present paper are all qualitatively different, moreover [CFU14; Ebr+13] do not give results for the hopcount or flooding time. Numerical solution of the maximization problem (4.2.4) for d = 2 yields the optimal (α˜ , β˜) pair to be approximately (0.8639, 1.500). The corresponding constant for the diameter is 2c /µ 0.8639 1.5 = 2 2 · · 1.668, which perfectly coincides with the one obtained in [CFU14] and [Ebr+13]. To the best of our knowledge no result has been proven before for thee flooding time.

4.3 Structure of RANs and EANs

4.3.1 Tree-like structure of RANs and EANs The construction method of RANs and EANs enables us to describe a natural way to code the vertices and active cliques of the graph parallel to each other. Let Σd := 1, 2, . . . , d + 1 be the symbols of the alphabet. In a d-dimensional simplex, the spher- { } ical vertex ﬁgure at a vertex v is the subspace of the simplex "cut out" by a small enough d-dimensional ball centered at v. The coding in two dimensions is illustrated in Fig- ures 4.1 and 4.2. In d-dimensions it is done by induction:

1. Initialization: Each vertex of the initial d-dimensional simplex is given a different auxiliary code i for i Σ . For each vertex i , the spherical vertex ﬁgure at 0 0 ∈ d 0 0 0i0 is given the label i. Furthermore, the single active clique (the interior of the simplex) gets the code O. See left-hand side of Figure 4.1. 94 Chapter 4. Distances in random and evolving Apollonian networks

030 030 030

3 3 3

O 1 O 2 2 1 O 2 2 1 3 1 3

3 31 1 3 2 32 3 2 1 2 1 2 33 1 020 010 020 010 020 010

FIGURE 4.1: Initial steps of coding RANs. Green codes represent active cliques, blue codes represent inactive cliques, and the red labels correspond to spherical vertex ﬁgures.

2. Step 0: Active clique O becomes inactive and O gets connected to all the vertices 0i0, thus creating the new active d-simplexes. The newly formed spherical vertex figures at each vertex i inherit the label i. The d 1-dimensional hy- 0 0 − perplanes defined by the new edges divide the d-dimensional ball centered at O into d + 1 spherical vertex figures. Each of these spherical vertex figures is given a different, unique label from Σd so that in every active clique, the d + 1 spherical vertex figures all have different labels from Σd. An active clique is assigned the code i corresponding to the label i of the spherical vertex figure inside the clique at the newly added vertex O. See center of Figure 4.1.

3. Step 1: One of the active cliques becomes inactive (in Figure 4.1 it is3) and we assign the newly added vertex v the code v of the clique that becomes inactive. The new spherical vertex ﬁgures around v are assigned different, unique labels from Σd same way as before. Then the new active cliques are assigned codes vj, j = 1, . . . , d + 1 (here vj means concatenation), where j corresponds to the label j of the spherical vertex ﬁgure inside the new clique at the vertex v. See right-hand side of Figure 4.1.

4. Induction step: Same as Step 1. An active clique u becomes inactive, the spherical vertex figures around u are assigned labels from Σd. Based on these labels, the newly added active cliques are given the codes uj, j Σ . See Figure 4.2. ∈ d Thus, at the beginning and end of each step, every (in)active vertex has a well- defined code and each spherical vertex figure has a well-defined label.

T3u T3u

3 3

u 1 u 2 u2 u1 3

2 1 2 1 T2u T1u T2u T1u

FIGURE 4.2: The general induction step in the coding of RANs when an active clique u is chosen. The neighbors of u are T u for i Σ . i ∈ d 4.3. Structure of RANs and EANs 95

Hence, each vertex in the graph has a code that is a concatenation of symbols from Σ . For a vertex u we write u = u u ... u for its code for some ` N, and we call d 1 2 ` ∈ the length of a code u = ` the generation of the vertex u. We deﬁne 1 = ... = d + | | |0 0| |0 1 = O = 0. For two vertices u and v with codes u = u u ... u and v = v v ... v , 0| | | 1 2 n 1 2 m respectively, we say that u is an ancestor of v if n m and u u ... u = v v ... v . ≤ 1 2 n 1 2 n We denote the latest common ancestor of u and v by u v and its code by u v, thus ∧ ∧ u v = min k : u = v . For codes u = u ... u and v = v ... v we denote | ∧ | { k+1 6 k+1} 1 n 1 m the concatenation u1 ... unv1 ... vm by uv and the corresponding vertex by uv. Furthermore, let (i) denote the index for which u(i) is the last occurrence of the symbol i Σ in u. For all i Σ we introduce the cut-operators ∈ d ∈ d

Tiu := u1 ... u(i) 1 and Piu = u(i) ... un. −

In the special case when u(i) = u1, then Tiu := O. Also, if there is no i in u or u = O, then we deﬁne Tiu :=0 i0. For future reference, we also deﬁne the operator Tminu, which gives the ancestor of u with length min T u . If there is at least one i Σ for i | i | ∈ d which T u = i or O, then T u := ∅ and T u := 0. i 0 0 min | min | Remark 4.3.1. Note that there is a one-to-one correspondence between the codes of length at most n and vertices of a rooted (d + 1)-ary tree of depth n. As a result, we use the codes u to denote vertices as well, that is, we identify vertices in a RAN or EAN with their codes and sometimes refer to u as a vertex. In this respect, the concept ‘u is an ancestor of v’ precisely means the ‘usual’ notion of being an ancestor: the unique path from v to the root in the (d + 1)-ary tree passes through u. Apart from these ‘tree’ edges, RANs and EANs have other edges as well. How- ever, we will see below that these extra edges always go upwards (or downwards) on a branch of a tree, hence the crucial tree-like properties of the structure are conserved. We collect the most important combinatorial observations in the following lemma. Lemma 4.3.2 (Tree-like properties of the coding). The coding of the vertices of a RAN or EAN described above has the following properties: (a) The d + 1 neighbors of a newly formed vertex with code u have codes T u, i Σ . i ∈ d Furthermore, for any edge with endpoints u and v either ‘u is an ancestor of v’ or vice versa.

(b) Any shortest path between two vertices with codes u and v must go through a vertex with code w which is an ancestor of u v or one of the initial vertices 1 ,..., d + 1 , O. ∧ 0 0 0 0 (c) For any two vertices with codes u and v,

Hop(u, v) (Hop(u, u v) + Hop(v, u v) 2. | − ∧ ∧ | ≤ Before the proof, let us interpret Lemma 4.3.2. Part (a) means that edges are only present between vertices along the same ancestral line. In particular, the ﬁrst d + 1 neighbours of a newly added vertex with code u can be determined by cutting off the last pieces of the code of u, up to the last occurrence of a given symbol i Σ . ∈ d The coding gives a natural grouping of the edges. Edges of the initial graph are not given any name. An edge is called a forward edge if its endpoints have codes of the form u and uj for j Σ . All other edges are called shortcut edges. So in a RAN at ∈ d each step one new forward edge and d shortcut edges are formed. Figure 4.3 below shows an example in two dimensions. Suppose at step n = 0 the ‘left’, ‘right’ and ‘bottom’ triangles were given the symbols 1, 2 and 3 respectively. 96 Chapter 4. Distances in random and evolving Apollonian networks

Then later each new vertex u with code u in the middle of a triangle gives rise to the new ‘left’, ‘right’ and ‘bottom’ triangles: to these we have to assign the codes v1, v2 and v3 respectively. On the left hand side is a planar embedding of the graph, while on the right the tree-like structure of the same graph becomes more apparent. Interpreting the initial graph as the root, the forward edges are the edges of the tree: along them we can go deeper down in the hierarchy of the graph. The shortcut edges only run along a tree branch: between vertices that are in the same line of descent, so we can ‘climb up’ to the root faster along these edges.

Coding of vertices grouping of edges O

u u = 132 u O v v = 3312 initial graph u v forward edges

shortcut edges v

FIGURE 4.3: Tree like structure of a realization of RAN2(8)

Part (b) is a consequence of part (a). It says that if we have two vertices with code u and v in the tree, then any shortest path between them must intersect a path from the initial graph to their latest common ancestor u v. Finally, part (c) says that up to ∧ a bounded additive error we can calculate the length of the shortest path by looking at the path from u v to u and v separately. ∧ Proof of Lemma 4.3.2. Part (a) is a direct corollary of the following observation: in an active simplex u, the code of the vertex whose spherical vertex figure is labeled by i is equal to Tiu. This is true for the initial Step 0. Thus it is enough to show that it is also true after the Induction step, refer to Figure 4.2. By construction, the label of the spherical vertex figure of u inside the active simplex ui is i, and of course Ti(ui) = u. By construction, the other spherical vertex figures inside the active simplex ui inherit the labels Σ i from the spherical vertex figures inside the simplex of u. The d \{ } induction hypothesis holds for u, thus the vertices of the simplex, in which u became inactive, have codes T u for j Σ i . The observation now follows because j ∈ d \{ } T (ui) = T u for all j Σ i . Finally, part (a) follows because a new inactive vertex j j ∈ d \{ } always gets the code of its active simplex. Part (b) follows from part (a): every vertex is connected to d + 1 vertices with code length shorter than u , and all these vertices are descendants of each other, i.e., they | | are in the path from u to the initial graph. The other vertices u is connected to are its descendants, i.e., of the form uw for some w. Hence, if we want to build a path from vertex u to v, we must go up in the tree to at least u v. ∧ Part (c). First observe that for all i Σ and all codes u, x, y ∈ d T (uxy) T (ux) , (4.3.1) | i | ≥ | i | i.e. the position of the last occurrence of a symbol in a code can not be earlier than that in some prefix of the same code. 4.3. Structure of RANs and EANs 97

We describe the shortest path u v with the help of Figure 4.4. The blue edges → give the shortest path from u to the initial graph. This path intersects the ancestral line of u v at the vertex z∅, keeping in mind that possibly z∅ = u v or z∅ is ∧ u u ∧ u already a vertex of the initial graph. For some non-empty word x, let (u v)x be the ∧ next vertex on the ∅ u shortest path below the level of u v. We deﬁne (u v)y → ∧ ∧ analogously on the shortest path ∅ v. →

u (u v)x ∧ (u v) RAN (0) ∧ d zv zu zu∅ v (u v)y ∧

FIGURE 4.4: Shortest path from u to v. Blue edges are on the shortest path from u, v to the initial graph, green edges are other edges present in the graph.

Let z be the vertex where the shortest path u v intersects the ancestral line of u → u v from (u v)x, thus z is somewhere in between u v and z∅ (could be equal to ∧ ∧ u ∧ u one of them). We deﬁne zv analogously from v and assume without loss of generality that z z (z and z could be the same). | u| ≤ | v| u v Observe that by part (a) there is an edge z z . Indeed, for some i Σ we have u − v ∈ d z = T ((u v)x). For this same i we also have T z = z because z is an ancestor u i ∧ i v u v of (u v)x. Moreover, with analogous reasoning, it follows that the four vertices ∧ u v, z , z , and z∅ form a complete graph (keeping in mind that potentially any two ∧ u v u of these vertices are the same). Thus, we can write the shortest path u v as → u (u v)x z z (u v)y v, → ∧ − u − v − ∧ → and also another u v path going through u v → ∧ u (u v)x z u v z (u v)y v, → ∧ − u − ∧ − v − ∧ → whose length is precisely Hop(u, u v) + Hop(v, u v). This is because there is no ∧ ∧ (u v)x u v edge unless u v = z . The assertion now follows. ∧ − ∧ ∧ u

4.3.2 Distances in RANs and EANs: the main idea With the help of the grouping of the edges as above, we can determine the distance between two arbitrary vertices u, v with codes u, v as follows: First, determine the generation of their latest common ancestor u v. Then deter- ∧ mine the length of their code below u v. Finally, determine how fast can we reach ∧ the latest common ancestor along the shortcut edges in these two branches, i.e., what is the minimal number of hops we need to go up from u and v to u v? ∧ If we pick u, v u.a.r., then we have to determine the typical length of codes in the tree and the typical number of shortcut edges needed to reach the typical common ancestor. If on the other hand we want to analyse the diameter or the ﬂooding time, we have to ﬁnd a ‘long’ branch with ‘many’ shortcut edges. Clearly, one can look at the vertex of 98 Chapter 4. Distances in random and evolving Apollonian networks maximal depth in the tree: but then - by an independence argument about the gs in the code and the length of the code - with high probability the code of the maximal depth vertex in the tree will show typical behaviour for the number of shortcut edges. On the other hand, we can calculate how many slightly shorter branches are there in the tree. Then, since there are many of them, it is more likely that one of them has a code with more shortcut edges needed than typical. Hence, we study the typical depth and also how many vertices are at larger, atypical depths of a branching process that arises from the forward edges of RANs. The effect of the shortcut edges on the distances is determined using renewal theory (also done in [AM08]) and large deviation theory. Finally, we optimize parameters such that we achieve the maximal distance by an entropy vs energy argument.

4.3.3 Combinatorial analysis of shortcut edges Now we investigate the effect of shortcut edges on this tree. Lemma 4.3.2 part (a) says that the shortcut edges of a vertex u in the tree lead exactly to Tiu, the prefixes of u achieved by chopping the code after the last occurrence of symbol i in the code. Recall that Piu = u(i) ... un denotes the postfix of u that starts with the last occurrence of the symbol i Σd in the code of u, while Tiu = u1 ... u(i) 1. ∈ − Moreover, recall the operator that gives the prefix with length min T u is T i | i | min and further, denote the length of the maximal cut by

AN Yd (u) := u Tminu = max Piu . (4.3.2) | | − | | i Σd {| |} ∈ This is the length of the maximal hop we can achieve from the vertex u towards the root in the tree via a shortcut edge. Consecutively using the operator Tmin we can decompose u into independent blocks, where each block, when reversed, ends at the ﬁrst position when all the symbols in Σd have appeared. We call such a block full coupon collector block. E.g. for u = 113213323122221131 this gives 1 132 1332 31222 21131. Let us denote the total | | | number of blocks needed in this decomposition by

k N(u) = max k + 1 : (Tmin) u = ∅ . (4.3.3) { 6 } Note that this is not the only way to decompose the code in such a way that we always cut only postﬁxes of the form P u: e.g. 1 132 1332 31222211 31 gives an alternative cut i | | | with the same number of blocks.

The following (deterministic) claim establishes that the decomposition along repet- itive use of Tmin (longest possible hops) is optimal.

Claim 4.3.3. Suppose we have an arbitrary code u of length n with symbol from Σd, that we want to decompose into blocks in such a way that from right to left, each block ends at the first appearance of some symbol in that block. Then, the minimal number of blocks needed is given by N(u). Proof. Consider two different decompositions of u into blocks: in the first decomposition use the operator Tmin consecutively, while in the second one we suppose that at least one block is not a full coupon collector block. Without loss of generality we may assume that this is the first block from the end of the code u. The endpoint of the first hop in the first decomposition is Tminu := Ti∗ u, while in the second decomposition the endpoint is T u for some j = i , with T u > T u . Hence, there is a w such j 6 ∗ | j | | min | that Tju = (Tminu)w. Conclude from (4.3.1) that Hop(Tminu, ∅) Hop(Tju, ∅): thus ≤ the number of blocks in the second decomposition can not be smaller than N(u). 4.3. Structure of RANs and EANs 99

AN Note that Yd ( ) and N( ) are deterministic operators when applied to a ﬁxed · · AN code u. Next we state the distributional properties of Yd (u) and N(u) when u is the code of a uniformly chosen active clique. The reason for the need of this is that both in the evolution of RAN and EAN, once a clique with code u becomes inactive and is replaced with the vertex with code u, the d + 1 new cliques that become active are exactly the direct descendants (children) of the vertex u in the d-ary tree. At each step in the evolution of the RAN, the clique to become inactive is chosen u.a.r. among the active cliques, and in the EAN an independent coin ﬂip with success probability qn (not depending on the code itself) determines for each active clique if it becomes inactive or stays active for the next step.

Claim 4.3.4. Let u and v be two active vertices chosen u.a.r. from a RANd(n) or EANd(n).

(a) Then, conditioned on the length of u, the symbols in Σd are distributed uniformly at each coordinate of u.

(b) Then, conditioned on the length of u and v, the symbols in Σd are independent and distributed uniformly at each coordinate after the u v + 1-th in both u and v. | ∧ | Proof. Assertion (a) follows from the symmetry of the dynamics. Assume there is a coordinate in which the symbols of Σd are not uniformly distributed, i.e. there is a symbol with greater probability than the others. If we permute the labels of Σd then another symbol has greater probability in that coordinate. However, the role of each symbol is symmetric due to the dynamics, therefore all symbols in that coordinate should also have the same probability, giving a contradiction. Due to the construction, two active cliques either share at most a hyperplane or one is contained in the other. In the former case, the evolution of the graph within the two cliques is independent, since the restriction of the uniform distribution to any subset is also uniform. This is the case with the two active cliques u1u2 ... u u v +1 | ∧ | and v1v2 ... v u v +1. Within these two cliques we can simply think that we start two independent| RANs.∧ | Conditioned on the lengths of u and v, it now follows from part (a) that the symbols in Σd are independent and distributed uniformly at each coordinate in the postﬁxes of u and v after the u v + 1-th coordinate. | ∧ | For every k 1, let us deﬁne the random variable ≥ ` (`) H := max ` : Y k , (4.3.4) k { ∑j=1 d ≤ } (`) where Yd are i.i.d copies of Yd in (4.1.2).

Lemma 4.3.5. Suppose u is a code of length k with symbols chosen u.a.r. from Σd at each position. Then d YAN (u) = min Y , k , d { d } d N(u) = Hk + 1. Proof. The last occurrence of any symbol i Σ in a uniform code is the ﬁrst occurrence ∈ d from backwards of the same symbol. Hence, reverse the code of u, and then P u is | i | the position of the ﬁrst occurrence of symbol i in a uniform sequence of symbols of length k, since u = k. Clearly P u = k if the symbol i does not occur in u. As | | | i | a result, P u has a geometric distribution with parameter 1/(d + 1) truncated at k. | i | Maximizing this over all i Σ we get the well-known coupon collector problem, that ∈ d has distribution Yd, truncated again at k. For the second part, since N(u) cuts down 100 Chapter 4. Distances in random and evolving Apollonian networks full coupon collector blocks from the end of the code of u consecutively, the maximal number of cuts possible is exactly the number of consecutive full coupon collector blocks in the reversed code of u, an i.i.d. code of length k. Since the length of each block has distribution min Y , k , and they are independent, the statement follows { d } by observing that the last, non-full block of the reversed code corresponds to the +1 in the statement.

2 Recall µd, σd from (4.1.3). From basic renewal theory [Fel68] the following central limit theorem holds as k ∞: → H k/µ d k − d (0, 1). (4.3.5) 2 3 −→ N kσd /µd q Furthermore, the expected value of Hk satisﬁes [Fel68, Chapter XII. 12. Problem 22.]

k 2σ2 + µ µ2 k E d d d [Hk] = + 2 − + o(1) = + O(1). (4.3.6) µd 2µd µd

4.4 Distances in RANs and EANs

In light of the main idea of the proof in Subsection 4.3.2 we begin with the analysis of the tree created by the forward edges of the graph.

4.4.1 A continuous time branching process There is a natural embedding of the evolution of the RAN into the evolution of a continuous time branching process (CTBP) [AN04], or a Bellman-Harris process. Namely, consider a CTBP where the offspring distribution is deterministic: each individual (equivalently, a vertex) has d + 1 children and the lifespan of each individual is i.i.d. exponential with mean one. Thus, after birth a vertex is active for the duration of its lifespan, then splits, becomes inactive and at that instant gives birth to its d + 1 offspring that become active for their i.i.d. Exp(1) lifespan. The process starts with a single individual that is called the root and who dies immediately at t = 0 giving birth to its d + 1 children. The bijection between the CTBP at the split times and a RANd is the following: the individuals that have already split in the CTBP are the vertices already present (inactive vertices) in the RANd, while the active (alive) individuals in the CTBP correspond to the active vertices (active cliques) in the RANd. This holds since at every step of a RANd, d + 1 new active cliques arise in place of the one which becomes inactive. Furthermore, in a RANd an active clique is chosen u.a.r. in each step which is – by the memoryless property of exponential variables – equivalent to the fact that the next individual to split in the CTBP is an active individual chosen u.a.r. We write GU(m) for the generation of a uniformly chosen active individual in the CTBP after m individuals have split, i.e., its graph distance from the root. The next two propositions describe the growth of our CTBP in terms of the typical size of GU(m) as well as the degree of relationship of two active individuals chosen u.a.r. in Proposition 4.4.1 and the maximal size of GU(m) together with its tail behaviour in Proposition 4.4.3. 4.4. Distances in RANs and EANs 101

Proposition 4.4.1. Let Z denote a standard normal random variable. Then as m ∞ → d+1 GU(m) log m d − d Z. (4.4.1) d+1 −→ d log m q Further, let GU, GV denote the generations of two active vertices chosen u.a.r. in the CTBP after the m-th split, and let us write GU V for the generation of the latest common ancestor of ∧ d U, V. Then the marginal distribution GU = GU(m), and

d+1 d+1 GU GU V d log m GV GU V d log m d − ∧ − , − ∧ − (Z, Z0), (4.4.2)  d+1 d+1  −→ d log m d log m  q q  where Z, Z0 are independent standard normal distributions. The proposition is an application of [Büh71, Theorems 2.5, 4.2] to the CTBP here with deterministic offspring distribution (d + 1 children). Before the proof, we need a lemma, that originates from Bühler [Büh71, Theorem 3.3], and the ﬁrst part can also be found e.g. in [BHH11]. First some notation: let us write Di, Si for the number of children of the i-th splitting vertex and the number of active individuals after the i-th split in a CTBP, and for an event A and random variable X let us write shortly P (A) := P(A D , S , i = 1, . . . , m), E [X] := E[X D , S , i = 1, . . . , m]. m | i i m | i i Claim 4.4.2. The generation GU(m) of an active individual U chosen u.a.r. after the m-th split in a CTBP satisﬁes the following indicator representation

m d Di GU(m) = ∑ 11i, where Em [11i] = , (4.4.3) i=1 Si and the indicators are independent conditioned on the sequence Di, Si, i = 1, . . . , m. Secondly, let us denote (U, V) a pair of individuals chosen u.a.r. after the m-th split. Let us further assume that the latest common ancestor U V of U and V reproduced at the τU V-th ∧ ∧ split. Then, conditioned on τU V the following two variables are independent and their joint distribution can be written as ∧

m m d 1 1 (GU GU V, GV GU V) = i, i0 , (4.4.4) − ∧ − ∧ ∑ ∑ i=τU V i=τU V ! ∧ ∧ where 1 1 Di Si Di Pm ( i, i0) = (1, 0) τU V < i = − | ∧ S S 1 i i − 1 1 Di Si Di Pm ( i, i0) = (0, 1) τU V < i = − (4.4.5) | ∧ Si Si 1 − 1 1 Di(Di 1) Pm ( i, i0) = (1, 1), τU V = i τU V i = − ∧ | ∧ ≤ Si(Si 1) − and conditioned on τU V, different indices are independent. ∧ Proof. A proof of the ﬁrst statement using the ancestral line can be found in [Büh71, Section 3.A] (see also Section 2.A for clearer explanations), but a proof based on induction can also be worked out. Here we give the core idea of the proof of Bühler. The ancestral line of an individual in a CTBP is the unique path from the individual to the root. For the time interval between the i-th and i + 1-th split we can allocate a unique 102 Chapter 4. Distances in random and evolving Apollonian networks individual on the ancestral line that was active in this time interval. For the following 1 1 1 observations, we condition on Di, Si, i = 1, . . . , m. Then Gm = 1 + 2 + + m, 1 1 ··· where the indicators i are conditionally independent and i = 1 if and only if the ancestor that was alive in the time interval between the i-th and i + 1-th split was newborn (born at the i-th split). Recall that the individual that splits at the i-th step is chosen u.a.r., as well as U is also chosen u.a.r. among the Sm many active individuals after the m-th split. Since in the interval between the i-th and (i + 1)-th split there were exactly Di many individuals newborn, and Si many alive, and the ancestor of U is equally likely to be any of them, this yields the probability P(1 = 1 D , S ) = D /S . i | i i i i The proof of the second statement follows from [Büh71, Section 3.B] in a similar manner: after time τU V, we write GU GU V as sums of indicators, where 11i is 1 if and ∧ − ∧ only if the individual alive between the i-th and (i + 1)-th on the ancestral line of U is newborn (born at the ith split). We do the same for GV GU V using 110s. Conditional − ∧ i on Di, Si, i = 1, . . . , m the pairs (11i, 11i0) become independent and their joint distribution is the one given in (4.4.5), since at each step, each pair of active individuals is equally likely to be the ancestors of U and V, and the ancestral lines merge precisely when the ancestors of U and V are two children of the vertex that splits at step i, giving the last line of (4.4.5).

Proof of Proposition 4.4.1. The proposition follows from Claim 4.4.2. More precisely, we note that in our case Di = d + 1 and Si = di + 1 are deterministic, hence

m d d + 1 GU(m) = ∑ 11i, where P (11i = 1) = , (4.4.6) i=1 di + 1

From this identity the expectation and variance of GU follows:

d + 1 2 1 E [G (m)] = log m + O(1), D [G (m)] = E [G ] + O m− . (4.4.7) U d U U The central limit theorem (4.4.1) holds for the standardization of GU(m) since the collection of Bernoulli random variables 11 m , m = 1, 2, . . . satisﬁes Lindeberg’s { i}i=1 condition. d For the second statement, GU = GU(m) is obvious by noting that the marginal of a uniformly chosen pair of vertices is a uniformly chosen vertex. Next, note that the 1 1 event ( i, i0) = (1, 1) means that the ancestral lines of U and V merge at the i-th split. To see that τU V has a limiting distribution we can use the following: ∧ m P(τU V k) = (1 P(τU V = i τU V i)) , (4.4.8) ∧ ∏ ∧ ∧ ≤ i=k+1 − | ≤ where the factors on the right hand side are the probabilities that the two ancestral lines do not merge at the i-th split. This tends to a proper limiting distribution since by (4.4.5) ∞ ∞ d + 1 P(τU V = i τU V i) = < ∞. ∑ ∧ ∧ ∑ i=1 | ≤ i=1 (di + 1)i

Hence, τU V has a limiting distribution, i.e. the limit limm ∞ P(τU V k) exists for ∧ α α→ ∧ ≤ every k. Choosing k = m we have limm ∞ P(τU V m ) = 1 for any α > 0, thus → ∧ ≤ log τU V / log m 0 in probability. Note that GU V also has a limiting distribution, ∧ → ∧ independent of m, since GU V is the generation of the individual that splits at the ∧ τU V-th split. ∧ 4.4. Distances in RANs and EANs 103

From here, one can show the joint convergence of (4.4.4) using Lindeberg CLT for m ( 1 + 1 ) linear combinations of ∑i=τU V +1 α i β i0 and get that the two variables in (4.4.2) tend jointly to a two-dimensional∧ standard normal variable.

Recall the deﬁnition of the function fd(c) from (4.1.5) and the constant c˜d that satisﬁes c˜ > (d + 1)/d, f (c˜ ) = 1. We will need the next proposition in the proof d d d − of Theorem 4.2.3.

Proposition 4.4.3. The exact asymptotic tail behaviour of GU(m) is given by

log (P (GU(m) > c log m)) lim = fd(c). (4.4.9) m ∞ m → log Further, after m splits the deepest branch in the CTBP satisﬁes

maxi m GU(i) P ≤ c˜ . (4.4.10) log m −→ d

θG (m)/ log m Proof. Let Λm(θ) := log E e U . Using (4.4.6) elementary calculation yields: h i m d+1 Λ (θ log m) = log 1 + (eθ 1) . m ∑ di + 1 − i=1 Hence, from the series expansion of log(1 + x) we can see that

1 d + 1 θ lim Λm(θ log m) = (e 1), m ∞ m d → log − which is the cumulant generating function of a ξ = Poi((d + 1)/d) random variable. The rate function of such a random variable is f (c). Hence, the conditions of the − d Gärtner-Ellis theorem [DZ10, Subsection 2.3] are satisﬁed, which implies (4.4.9). Our CTBP is a special case of so-called random lopsided trees [CG01; KR89]. The maximal depth of such trees was studied by Broutin and Devroye [BD06] in a more general framework. Thus (4.4.10) is just an application of [BD06, Theorem 5 and Re- mark afterwards] with our notation.

Remark 4.4.4. To see that c˜d should be the right constant in (4.4.10) we can argue that from (4.4.9) it follows that the sum ∑m P (GU(m) > c log m) < ∞ for any c > c˜d. Thus by the Borel-Cantelli lemma, for any such c there are only ﬁnitely many m such that the event G (m) > c log m holds, giving the whp upper bound c˜ log m on the { U } d depth of the CTBP.

4.4.2 Proof of Theorem 4.2.1 and 4.2.5

Proof of Theorem 4.2.1. Pick a pair of active vertices u, v u.a.r. from a RANd(n). We write u , v for their generation. As before, we write u v for their latest common | | | | ∧ ancestor, i.e., the longest common prefix of their codes. Let us define the distinct postfixes u, v after u v by ∧ u =: (u v)u, v =: (u v)v. e e ∧ ∧ e e 104 Chapter 4. Distances in random and evolving Apollonian networks

By Lemma 4.3.2 (c) and Claim 4.3.3 the length of the shortest paths between u, v (up to a small additive error, which cancels in the limit) satisﬁes

dist(u, v) = N(u) + N(v), and Proposition 4.4.1 describes the typical distancee betweene u and v along the tree (i.e., only using forward edges and no shortcut edges). Since u and v were chosen uniformly at random among the active vertices after n splits. Hence, we can write d d u = G (n), v = G (n), where U, V denotes two uniformly chosen alive individuals | | U | | V in the CTBP in Section 4.4.1. By the same reasoning (and dropping the dependence of n for shorter notation),

d d d u v = GU V, u = GU GU V, v = GV GU V. (4.4.11) | ∧ | ∧ | | − ∧ | | − ∧ Further, by Claim 4.3.4, the symbolse in the codeseu and v are i.i.d. uniform on Σd (after the ﬁrst symbol in both of the codes, which has to be different by the deﬁnition of u v). Hence, by Lemma 4.3.5, ∧ e e d dist(u, v) = HGU GU V + HGV GU V . (4.4.12) − ∧ − ∧ Using (4.3.6) we have

E [GU GU V ] E [HGU GU V ] = E [E [HGU GU V GU GU V ]] = − ∧ + O(1). − ∧ − ∧ | − ∧ µd

Furthermore, (4.4.7) and the fact from the proof of Claim 4.4.2 that GU V has a limiting ∧ distribution implies that for any a +∞ sequence m → 1 d + 1 E [HGU GU V ] = log m + O(am). − ∧ µd d

To obtain a central limit theorem for HGU GU V observe that − ∧ 1 d+1 1 HG G log m HG G (GU GU V ) U U V µd d U U V µd GU GU V − ∧ − = − ∧ − − ∧ ∧ d+−1 d+1 2 3 2 3 · s log m d log m σd /µd (GU GU V )σd /µd d − ∧ (4.4.13) q 1 q 1 d+1 µ (GU GU V ) µ d log m + d − ∧ − d . d+1 2 3 d log m σd /µd q The ﬁrst factor on the right hand side, conditionally on GU GU V with GU GU V − ∧ − ∧ → ∞, tends to a standard normal random variable independent of GU by the renewal CLT in (4.3.5) and the second factor tends to one in probability by (4.4.2). By (4.4.2) again, the second term tends to a (0, µ /σ2). Since the length of the codes u, v are N d d independent of the symbols in these codes, HGU GU V GU GU V is independent of − ∧ | − ∧ GU GU V. As a result, the two limiting normals arising from the two summandse e on − ∧ the right hand side of (4.4.13) are also independent, thus

1 d+1 HGU GU V µ d log m d − ∧ − d 2 (0, 1 + µd/σd ). (4.4.14) d+1 2 3 −→ N d log m σd /µd q 4.4. Distances in RANs and EANs 105

By conditioning ﬁrst on GU V, (as in the proof of Proposition 4.4.1) and using that the ∧ symbols in the code of u, v are all i.i.d. uniform in Σd, one can show that (HGU GU V , HGV GU V ) − ∧ − ∧ tend jointly to two independent copies of (0, 1 + µ /σ2) variables. By (4.4.12) it fol- N d d lows that Hop(n) = HeGUe GU V + HGV GU V , the ﬁrst statement of the Theorem 4.2.1 − ∧ − ∧ immediately follows by normalising such that the total variance is 1. The second statement follows by calculating how many active cliques a vertex with degree k is contained in: a vertex with degree d + 1 is contained in d + 1 cliques, and when the degree of a vertex v increases by 1, then the number of cliques containing v increases by d 1, thus a vertex with degree k d + 1 is contained in exactly − ≥ Q = 2 + (k d)(d 1) (4.4.15) k − − active cliques. This means that the inactive vertex v is connected to exactly Qk many active vertices with an edge. It is clear that the total number of active vertices after n steps is A(n) = dn + d + 1. This implies that choosing two inactive vertices x, y according to the size-biased distribution given in (4.2.2) is equivalent to choosing two active cliques U, V chosen u.a.r. that are neighbouring these vertices. The distance between x, y is then between N(u) + N(v) 2, N(u) + N(v) since by Lemma 4.3.2 − x = T u for some i Σ , hence we can gain at most 1 hop by considering x instead of i ∈ d the clique U and the same holds fore y andeV. Hence,e the CLTe for U, V implies a CLT for two vertices picked according to the probabilities in (4.2.2).

Remark 4.4.5. Let us denote the generation of the m-th splitting vertex by Gm. Since at each split in the CTBP exactly one new inactive vertex is created, namely, a uniformly d chosen active vertex becomes inactive, we have G = G (m 1). Hence, ifb we would m U − like to choose an inactive vertex of RANd(n) uniformly at random, then its distance from the root has distribution GU(X) where X is ab random variable uniform in the set 0, 1, . . . , n 1 , with G (0) = 1. With a similar argument than the one in Claim { − } U 4.4.2, one can obtain that the latest common ancestor of two inactive vertices chosen u.a.r. also has a limiting distribution, and if Hop(n) denotes the distance between d+1 P them, one can obtain Hop(n)/(2 d log n) 1. But it is also not hard to see that −→d the CLT does not hold anymore, (since it does not hold for GU(X) for X uniform in 0, 1, . . . , n 1 ). d { − } Proof of Theorem 4.2.5. The proof follows analogous lines to the proof of Theorem 4.2.1, hence we give only the sketch. The main idea here is that the tree can be viewed as a CTBP where at step i, each active individual splits with probability qi or stays active for the next step with probability 1 q . Hence, Proposition 4.4.1 can be modiﬁed − i as follows: the generation of an active individual picked u.a.r. after the m-th split satisﬁes m d 1 GU(m) = ∑ i, i=1 1 e where i = 1 if and only if the individual on thee ancestral line of U is newborn at the i-th step. Note that in this case, the indicators are independent even without condi- 1 tioning,e and P( i = 1) = qi, since at each step each individual splits with the same probability, independently of each other. Since splitting happens with probability qi at step i, the CLTe for GU(m) holds by Lindeberg CLT. Now, for two individuals U, V picked u.a.r., with GU V, τU V as in Proposition 4.4.1, we have ∧e ∧ m m d 1 1 (GU GU V, GV GU V ) = i, i0 , − ∧ − ∧ ∑ ∑ i=τU V i=τU V ! ∧ ∧ e e 106 Chapter 4. Distances in random and evolving Apollonian networks

1 1 where different indices are independent and conditioned on τU V, i, 0 are indepen- ∧ i dent indicators with P(1 = 1) = P(1 = 1) = q . Since the variance ∑ q (1 q ) i i0 i i i − i → ∞, the joint CLT follows in a similar manner then for Proposition 4.4.1e eif we can show that τU V has a limitinge distribution.e For this note that similarly as in (4.4.8), ∧ m P(τU V k) = (1 P (τU V = i τU V i)) , (4.4.16) ∧ ∏ ∧ ∧ ≤ i=k+1 − | ≤ and the factors on the right hand side express that the two ancestral lines of U, V do not merge yet at step i. Let us write Ai for the number of active vertices at step i. Then at step i there are Zi := Bin(Ai, qi) many vertices that split, each of them producing d + 1 new active vertices, and hence the probability that the two ancestral lines merge at step i, conditioned on Ai, Ai+1 equals

Zi (d + 1)d P (τU V = i τU V i, Ai, Ai+1) = · , (4.4.17) ∧ | ∧ ≤ A (A 1) i+1 i+1 − where Ai+1 = Ai + d Zi, the new number of active vertices after the ith split. We obtained the rhs of (4.4.17) by observing that if U, V was chosen uniformly at random, each pair of individuals at step i, A (A 1)/2 in total, is equally likely to be the i+1 i+1 − ancestors of them, and there are Zi(d + 1)d/2 many pairs that make the ancestral lines merge. If the sum in i N on the right hand side of (4.4.17) is a.s. ﬁnite then (4.4.16) ∈ ensures that τU V has a proper limiting distribution. Hence we aim to show that this ∧ is the case whenever the total number of inactive vertices N(n) ∞, i.e., → ∞ Z (d + 1)d i < ∞ a.s. on N(n) ∞ . ∑ A (A 1) { → } i=1 i+1 i+1 − Since A = N(i + 1)d + d + 1, and Z = N(i + 1) N(i), we can approximate the i+1 i − above sum by ∞ d(N(i + 1) N(i)) (d + 1) ∑ − 2 . i=1 (dN(i + 1)) Now we can interpolate N(i) with a continuous function and then this sum is almost surely ﬁnite if and only if

T N0(x) 1 1 lim 2 dx = lim T ∞ 1 N(x) T ∞ N(T) − N(1) → Z → is almost surely ﬁnite. In particular, this holds when N(n) ∞. Further, as long → as ∑n N qn = ∞, N(n) ∞ holds a.s. by the second Borel-Cantelli lemma: in each ∈ → step we add at least a new vertex with probability qn. The CLT then for the distances follows in the exact same manner as in the proof of Theorem 4.2.1.

4.4.3 Proof of Theorem 4.2.3 We need some preliminary statements before the proof. Recall from (4.1.4) the definition of the large deviation rate function Id(x) of Yd and also Hk as the number of consecutive occurrences of full coupon collector blocks in a code of length k from (4.3.4). 4.4. Distances in RANs and EANs 107

Lemma 4.4.6. For 1 β µ /(d + 1),H satisﬁes the large deviation ≤ ≤ d k

1 β β µd lim log P Hk > k = Id . (4.4.18) k ∞ k µd − µd β → (i) Proof. Let Yd be i.i.d. distributed according to Yd. Since

β kβ/µd β µ P P (i) d Hk > k = ∑ Yd < k , µd i=1 µd · β we can apply Cramér’s theorem [DZ10, Subsection 2.2] to obtain (4.4.18).

We have seen at the end of the proof of Theorem 4.2.1 that switching from inactive vertices to neighbouring active vertices/cliques only changes the distances by at most 2, hence we rather investigate the diameter of the graph by active vertices. Let us denote the set of active vertices at step n by . We index by vertices u and An An denote one picked u.a.r by U. We have seen that = dn + d + 1. Our aim is to |An| estimate the expected number of u with distance at least xc˜ log n/µ from the ∈ An d d root, for some x 1. ≥ Recall the deﬁnition of the function g(α, β) from (4.1.6), and let us deﬁne for an x 1 ≥ (α(x), β(x)) := arg sup g(α, β) : αβ = x . (4.4.19) α,β { }

Claim 4.4.7. For any x 1, deﬁne the indicator variables for each vertex u ≥ ∈ An

Ju(x) := 11 N(u) > xc˜d log n/µd .

Then, with (α(x), β(x)) as in (4.4.19),

1 lim log E Ju(x) = g(α(x), β(x)). (4.4.20) n ∞ log n ∑ → u n h ∈A i Proof. Note that

1 E J (x) = (dn + d + )P H xc˜ n µ n ∑ u 1 GU (n) d log / d , (4.4.21) |A | n u ≥ h |A | ∈An i d where we used Lemma 4.3.5 for the distributional identity N(u) = HGU (n) for a uniformly chosen u (see also the argument above (4.4.11)). ∈ An x β(x) P P P (HGU (n) c˜d log n) (GU(n) > α(x)c˜d log n) Hα(x)c˜d log n > α(x)c˜d log n , ≥ µd ≥ µd where we used that x = α(x)β(x) and that the symbols in a uniformly chosen u are i.i.d. uniform in Σd, and Hk is increasing in k. Finally, multiplying both sides by dn + d + 1, taking the logarithm, and dividing by log n, applying (4.4.9) and (4.4.18) (with k = α(x)cd log n and only dividing by log n instead of α(x)cd log n) we arrive at

log E ∑u Ju(x) ∈An β(x) lim 1 + fd(α(x)c˜d) α(x)c˜d Id(µd/β(x)) = g(α(x), β(x)). n ∞ h n i → log ≥ − µd 108 Chapter 4. Distances in random and evolving Apollonian networks

For the upper bound, let us ﬁx a small ε > 0, and set

i?(ε) := max i : (α(x) (i + 1)ε)c˜ (d + 1)/d , { − d ≥ } i (ε) := min i : (α(x) iε)c˜ xc˜ ? { − d ≥ d} Then we can decompose the event H xc˜ log n/µ according to which ε- { GU (n) ≥ d d} length interval GU(n)/(c˜d log n) falls into, and use the monotonicity of Hk in k to get x P P (HGU (n) c˜d log n) H(d+1) log n/d xc˜d log n/µd ≥ µd ≤ ≥ + P(GU(n) > xc˜d log n) i?(ε) + P H(α(x) iε)c˜ log n xc˜d log n/µd ∑ − d ≥ i=i?(ε) P((α(x) (i + 1)ε)c˜d log n < GU(n) < (α(x) iε)c˜d log n) · − − (4.4.22) For the ﬁrst term, we use (4.4.18) to see that

log(P H(d+1) log n/d xc˜d log n/µd ) xc˜d (d + 1)µd lim ≥ = Id . (4.4.23) n ∞ log n − µ xc˜ d → d d For the i-th summand in the third term, we use an upper bound by dropping the upper restriction on GU(n), use (4.4.18) again and also (4.4.9) to get

log P H(α(x) iε)c˜ log n xc˜d log n/µd P((α(x) (i + 1)ε)c˜d log n < GU(n)) − d ≥ · − xc˜ µ (α iε) log n d I ( d − ) + f (α(x) (i + 1)ε)c˜ (1 + o(1)), ≤ · − µ d x d − d d (4.4.24) where the (1 + o(1)) disappears when dividing by log m and taking the limit as n → ∞. The second term can be treated similarly, except that there there is no part coming from the LDP of the H .-s. This is not surprising since this is the point where the · length of the code becomes so large that a typical number of shortcut edges already exceeds xc˜d log n/µd. To ﬁnish the upper bound, note that setting i = i?(ε) + 1, the rhs of (4.4.24) exactly gives the rhs of (4.4.23), while setting i = i?(ε) yields the second term, since in this case the rate function I ( ) vanishes. Further, note that the terms in (4.4.22) are addi- d · tive. This implies that when taking logarithm and dividing by log n, the largest term will dominate and determine the leading exponent. As a result,

P(H (n) xc˜ log n/µ ) lim GU ≥ d d n ∞ n → log xc˜d µd(α(x) iε) max Id( − ) + fd (α(x) (i + 1)ε)c˜d . ? ≤ i [i?(ε),i (ε)+1] − µd x − } ∈ To ﬁnish, let ε 0, and note that the ith term on the rhs is g(z, x/z) 1 for some → − z R. Since the maximum of this expression is taken at z = α(x), this ﬁnishes the ∈ proof.

Claim 4.4.8 (Monotonicity of g(α(x), β(x)) in x). The function g(α(x), β(x)) is continuous and strictly monoton decreasing for x > (d + 1)/(dc˜d). 4.4. Distances in RANs and EANs 109

c˜d µd Proof. Recall that g(α, β) := 1 + fd(αc˜d) αβ Id . The continuity follows from − µd β the fact that g(α, β) is differentiable. For the monotonicity, consider x1 > x2 > 1. We have to show that the maximum of the function g(α, β) on the hyperbola β = x1/α is smaller than that and on β = x2/α. Let g1 := g(α(x1), β(x1)), g2 := g(α(x2), β(x2)). Note that fd(αc˜d) < 0 and monoton decresing in α as long as α > (d + 1)/(dc˜d), while the second term αβI (µ /β) < 0 and monoton decreasing in β as long as β > 1. − d d Since x1 > (d + 1)/(dc˜d), at least one of the inequalities α(x1) > (d + 1)/(dc˜d) and β(x1) > 1 must hold. Suppose ﬁrst that α(x1) > (d + 1)/(dc˜d) holds. Then, if x2/β(x1) > (d + 1)/(dc˜d), then clearly g = g(α(x ), x /α(x )) < g(x /β(x ), β(x )) g and we are done. If 2 1 1 1 2 1 1 ≤ 1 on the other hand x2/β(x1) < (d + 1)/(dc˜d), then look at the point on the hyperbola (d + 1)/(dc˜d), x2/(d + 1)/(dc˜d). Since we decreased both coordinates, g1 < g((d + 1)/(dc˜d), x2/(d + 1)/(dc˜d)) holds as long as x2/(d + 1)/(dc˜d) > 1. This must hold since otherwise the whole hyperbola β = x /α would be in the region α < d + 2 { 1/(dc˜ ) β < 1 , which would mean that x < (d + 1)/(dc˜ ) which contradicts d } ∪ { } 2 d our original assumption. If β(x1) > 1, then the argument is similar by ﬁrst decreasing β to x2/α(x1) or to 1 (whichever is larger), and in case we had to decrease it to 1 than we further decrease α(x1) to x2 and again using that x2 > 1 implies that x2 > (d + 1)/dc˜d.

Proof of Theorem 4.2.3. First, we wish to choose the largest possible x in Ju(x) so that limn ∞ P ∑u Ju(x) > 0 > 0, that is, there is at least one active clique that has → n distance xc˜ log∈An/µ from the root. Note that if x is such that g(α(x), β(x)) < 0, then d d by Claim 4.4.7 and Markov’s inequality we have

g(α(x),β(x))(1+o(1)) P ∑ Ju(x) > 0 E ∑ Ju(x) = n 0. (4.4.25) u ≤ u → ∈Am h ∈An i Thus necessarily x has to have g(α(x), β(x)) 0. Next we work out the lower bound. ≥ For this, we shall need a upper bound on the second moment

2 E ∑ Ju(x) = E[ ∑ Ju(x) + ∑ E[Ju(x)Jv(x)] u n u n u,v ,u=v h ∈A i ∈A i ∈An 6 Note that the second term equals

2 (dn + d + 1) P(HGU > xc˜d log n/µd, HGV > xc˜d log n/µd)

where HGU and HGV is the minimal number of hops needed to reach the root from two active vertices chosen independently and u.a.r.. As before, let us write U V for ∧ the latest common ancestor of U and V. Then we can write

P(HGU > xc˜d log n/µd, HGV > xc˜d log n/µd)

=P(HGU V + HGU GU V > xc˜d log n/µd, HGU V + HGV GU V > xc˜d log n/µd). ∧ − ∧ ∧ − ∧ Pick any function ω(n) ∞ that also satisﬁes ω(n) = o(log n) (for instance, ω(n) = → log log n will do), then We can bound the right hand side from above as follows: x x P(HGU V > ω(n)) + P(HGU GU V > c˜d log n ω(n), HGV GU V > c˜d log n ω(n)) ∧ − ∧ µd − − ∧ µd − (4.4.26) Using the proof of Claim 4.4.2, we know that the joint distribution of two active individuals U, V picked u.a.r. satisﬁes that their common ancestor GU V has a limiting ∧ 110 Chapter 4. Distances in random and evolving Apollonian networks distribution. Hence, for any ω(n) ∞, →

P(HGU V > ω(n)) 0. (4.4.27) ∧ →

Further, conditioned on the splitting time τu v of U V, the joint distribution of GU ∧ ∧ − GU V, GV GU V can be described as the sum of indicators, see (4.4.4). Further, the ∧ − ∧ two sums are asymptotically independent, and also the symbols in the code u, v of U and V are independent and uniform in Σ after u v, the code of U V. Hence, d ∧ ∧ choosing n large enough, the ω(n) term becomes negligible and we get that

P(HGU GU V > xc˜d log n/µd ω(n), HGV GU V > xc˜d log n/µd ω(n)) − ∧ − − ∧ − = P(HGU GU V > xc˜d log n/µd) P(HGV GU V > xc˜d log n/µd)(1 + o(1)) (4.4.28) − ∧ · − ∧ 2 = P(HGU > xc˜d log n/µd) (1 + o(1)), where the o(1) 0 as n ∞, and in the last equation we used again the fact that → → GU V has a limiting distribution. Combining (4.4.27) with (4.4.28) to bound (4.4.26), ∧ we arrive at

2 E ∑ Ju(x) u h ∈An i E 2 P 2 = ∑ Ju(x) + (nd + d + 1) (HGU > xc˜ log n/µd) (1 + o(1)) + o(1) u h ∈An i 2 = E ∑ Ju(x) +E ∑ Ju(x) (1 + o(1)). u u h ∈An i h ∈An i (4.4.29) From a Cauchy-Schwarz inequality followed by (4.4.29), we get

2 E[∑u Ju(x)] P J (x) > 0 ∈An ∑ u ≥ 2 u n E ∈A ∑u n Ju(x) ∈A (4.4.30) h i 2 E[∑u Ju(x)] ∈An , ≥ 2 E[∑u Ju(x) + E[∑u Ju(x)] (1 + o(1)) ∈An ∈An i and the right hand side is strictly positive in the limit as n ∞ if and only if g(α, β) → ≥ 0 (using Claim 4.4.7 again for each term on the rhs). From this and the monotonicity of g(α(x), β(x)) in x (see Claim 4.4.8 it is immediate that the largest diameter can be achieved when picking x := x˜ so that g(α(x˜), β(x˜)) = 0. Apply (4.4.25) with x = x˜(1 + ε) and (4.4.30) with x := x˜(1 ε) to ﬁnally conclude that as n ∞ − →

maxu n HGU (n) P c˜d ∈A x˜. (4.4.31) log n −→ µd

The statement of Theorem 4.2.3 for the ﬂooding time now follows from the fact that if U is an active clique picked u.a.r. after the n-th step of the evolution of the RAN, then

d Flood(n) = HGU GU V + max HGV GU V . − ∧ v (n) − ∧ ∈A Now, the proof of Theorem 4.2.1 (or Proposition 4.4.1) implies that the CLT holds for generation GU GU V, and since the symbols are uniform in the code of U, similarly − ∧ 4.4. Distances in RANs and EANs 111

as in (4.4.13), the CLT holds for HGU (n) as well. Further, since in Flood(u, v) we max- imise the distance over the choice of the other vertex V, clearly whp we can pick V such that the latest common ancestor U V is the root itself. This combined with the ∧ fact that the distance changes only by at most 2 if we consider active cliques instead of vertices in the graph implies and the statement of the theorem follows from the distri- d butional convergence of H / log n (d + 1)d/µ and (4.4.31). For the diameter GU (n) −→ d we have Diam(n) d maxu (n) HGu = 2 ∈A , log n log n since for any ε > 0, whp there are at least two vertices that are not closely related to cd each other and both satisfy HG / log n > x˜(1 ε), but whp there are no vertices u µd − cd that satisfy HG / log n > x˜(1 + ε). e v µd e We are left to analyse the maximization problem. First of all, it is elementary to see (e.g. using Claim 4.4.8 or elementary two-dimensional calculus) that solving (4.4.19) and then choosing x˜ so that g(α(x˜), β(x˜)) = 0 is equivalent to the maximization problem in (4.2.4). However, two dimensional techniques give a better understanding of the solution x˜ = α(x˜) β(x˜). For short we write α(x˜) := α˜ , β(x˜) := β˜. · Lemma 4.4.9. The maximization problem (4.2.4) has a unique solution (α˜ , β˜) (0, 1] µd ∈ × [1, d+1 ], and further this solution satisfies ˜ 1 d + 1 ˜ β µd 1 + fd(α˜ c˜d) α˜ = exp Id0 (µd/β) , Id = . c˜d d − µd β˜ α˜ c˜d Proof. Define the Lagrange multiplier function (α, β, λ) := αβ λg(α, β). Necessar- L − ily the optimal (α˜ , β˜) satisfies (α˜ , β˜, λ) = 0. The partial derivative (α, β, λ) = 0 ∇L L λ0 simply gives the condition g(α, β) = 0. Further, the optimising λ can be expressed e from (α, β, λ)α0 = 0 and (α, β, λ)0β = 0 and satisfies L L e β α λ = ∂ = ∂ . ∂α g(α, β) ∂β g(α, β) e β µd After differentiation of g(α, β) = 1 + fd(αc˜d) αc˜d Id , rearranging terms and − µd β using that f (x) = log d x we obtain the first condition. To check the sufficiency d0 − d+1 we look at the bordered Hessian 0 ∂g ∂g ∂g ∂g ∂α ∂β 0 ∂α ∂β 2 2 ∂g ∂ αβ ∂ αβ = ∂g  ∂α ∂α2 ∂α∂β   ∂α 0 1  . 2 2 ∂g ∂ αβ ∂ αβ ∂g 1 0  2  ∂β  ∂β ∂α∂β ∂β        2 Its determinant is ∂2g(α, β)/∂α∂β > 0, thus the condition is also sufficient. We note that the solution can be approximated by numerical methods. Remark 4.4.10. We mention here the difficulties in the analysis of the diameter and flooding time of EANs: the main difficulty here is to understand the proper corre- lation structure of the codes (and shortcut edges) on the vertices of the BP: (a) The 1 corresponding BP tree is fatter than the BP for RAN as soon as n− = o(qn). (b) In each step each vertex splits independently of the past with probability qn. (a) and (b) together imply that even though we do understand the marginal distribution of the 112 Chapter 4. Distances in random and evolving Apollonian networks

symbols of a clique U picked u.a.r. is uniform in Σd, still it is more likely that the ’neighbouring codes’ are also present in the graph and hence codes for which N(u) is large are more likely to appear. Hence we expect that the diameter will have a larger constant in front of ∑ qi than the constant in front of log n for RAN. (Com- pare it to the diameter of the deterministic AN: with q 1 it is not hard to see that n ≡ Diam(ANd(n)) = 2n/(d + 1)). 113

Appendix A

Basic dimension theoretic deﬁnitions

Here we collect the various types of fractal dimensions used in the thesis. For further reference, see any of the excellent books [Fal85; Fal90; Fal97; Mat95]. Let F be a subset of Rd and s 0. For any δ > 0 we deﬁne ≥

s s δ(F) = inf ∑ Ui : F Ui, Ui δ , (A.0.1) H ( | | ⊆ | | ≤ ) i [i where U denotes the diameter of U. Any such collection Ui in the definition is | | { s} s called a δ-cover of F. The s-dimensional Hausdorff measure of F is (F) = limδ 0 (F). H → Hδ The limit exists (can be 0 or infinity), because the infimum increases. The Hausdorff dimension of F is

dim F = inf s 0 : s(F) = 0 = sup s 0 : s(F) = ∞ . (A.0.2) H { ≥ H } { ≥ H } Hausdorff dimension has many favorable properties, though usually it is hard to calculate. On the other hand box-counting dimension, also commonly called Minokwski-dimension, is easier to calculate or estimate, but it has serious drawbacks. For example countable sets can have dimension strictly larger than zero. The lower and upper box dimensions of a set F are deﬁned by

log Nδ(F) log Nδ(F) dimBF = lim inf and dimBF = lim sup , (A.0.3) δ 0 log δ δ 0 log δ → − → − respectively, where Nδ(F) is the smallest number of sets required for a δ-cover of F. Nδ can be replaced with various other definitions based on covering or packing F at scale δ, see [Fal90, Section 3.1]. We shall always use what is most convenient for us. If the limit exists, i.e. dimBF = dimBF, then this common value is called the box dimension of F, denoted dimB F. We do not define packing dimension using packing measures, because we will not use it directly. Instead we mention that there is an equivalent definition using the upper box dimension. Namely

∞ dimP F = inf sup dimBFi : F Fi , ( i ⊆ ) i[=1 where the inﬁmum is taken over all countable partitions F of F. This has the fol- { i} lowing consequence [Fal90, Corollary 3.9], which we will use later. 114 Appendix A. Basic dimension theoretic deﬁnitions

Proposition A.0.1. Let F Rd be a compact set such that every open set intersecting F ⊂ contains a bi-Lipshitz image of F, we get that dimP F = dimBF. The last dimension we define is the affinity dimension, introduced by Falconer [Fal88a]. The singular values 0 < α ... α of a non-singular d d matrix A are the pos- d ≤ ≤ 1 × itive square roots of the eigenvalues of AT A, where AT is the transpose of A. The geometric interpretation of the singular values is that the linear map x Ax maps 7→ the unit disk to an ellipse with principal semi-axes of length equal to the singular values. Hence, roughly speaking, the singular values indicate how much the map contracts (or expands) in different directions. For s [0, d] define the singular value ∈ function (introduced in [Fal88a])

s s s +1 −d e φ (A) = α1α2 ... α s 1α s . (A.0.4) d e− d e Given a ﬁnite collection of contracting matrices the afﬁnity dimension is A ∞ s = s ( ) = inf s : φs(A ... A ) < ∞ . (A.0.5) A A A ∑ ∑ i1 · · ik ( k=1 i1,...,ik )

When the matrices define a self-affine set Λ, we use the notation dimA Λ. Now we define the local dimension and Hausdorff dimension of a measure.

Deﬁnition A.0.2. Let µ be a Borel probability measure on Rd and x spt(µ). We deﬁne the ∈ local dimension of the measure µ at x by

log µ(B(x, r)) dµ(x) := lim , r 0 log r → if the limit exists. Otherwise we take lim inf and lim sup instead of lim and we obtain the lower local dimension dµ(x) and the upper local dimension dµ(x), respectively. The measure µ is exact dimensional if for µ-almost every x the limit exists and is equal to a constant.

Deﬁnition A.0.3. Let µ be a mass distribution. The upper and lower Hausdorff dimension of µ are deﬁned

dim µ := inf dim E : µ(Ec) = 0 , H { H } dim µ := inf dim E : E is a Borel set with µ(E) > 0 . H { H } When µ is exact dimensional, then the quantities are equal, simply called the Hausdorff dimension of µ. This is the case throughout the thesis. 115

Appendix B

No Dimension Drop is equivalent to Weak Almost Unique Coding

In this appendix, we prove that for self-similar IFSs on the line and Bernoulli measures the separation conditions No Dimension Drop (NDD) and Weak Almost Unique Coding (WAUC) are equivalent. We recall notation and deﬁnitions.

Notation Let = h (x) := r x + u M be a contractive self-similar IFS on the real line with H { ıˆ ıˆ ıˆ}ıˆ=1 attractor Λ . The symbolic space is Σ = 1, 2, . . . , M N and the natural projection H H { } is Π (ıˆ) := limn ∞ hıˆ n(0) for ıˆ Σ . Deﬁne a partition of Σ by H → | ∈ H H 1 ξ(ıˆ) := Π− Π (ıˆ). H H As we noted earlier in this paper, ξ is a measurable partition of Σ. We write ξ for the σ-algebra generated by the measurable partition ξ. Then the elements of ξ are unions of the elements of ξ. For a probability vector q = (q1,..., qM) we denote the Bernoullib measure on Σ by µq. Then there exists a Σ Σ , with µq(Σ ) = 1 suchb that for H H ⊂ H H all ıˆ Σ there existsb a probability measure µξ(ıˆ) deﬁned on ξ(ıˆ) such that ∈ H b b Forb all A Σ Borel set the mapping ıˆ µξ(ıˆ)(A) is ξ-measurable and • ⊂ 7→ for all Borel sets U Σ we have • ⊂ H b

µq(U) = µξ(ıˆ)(U)dµq(ıˆ). (B.0.1) Z

The push forward measure νq = (Π ) µq is the self-similar measure with support ∗ Λ . The entropy and Lyapunov exponentH of the system are H M hµq = log q q and χνq = ∑ qıˆ log rıˆ = log r q, − h i − ıˆ=1 − h i

M qi respectively, where c q = ∏ c . Now we recall two separation conditions from h i ıˆ=1 i Deﬁnition 2.1.9. 116 Appendix B. No Dimension Drop is equivalent to Weak Almost Unique Coding

Deﬁnitions We say that has No Dimension Drop (NDD) if for all probability vectors q with H strictly positive entries we have

hµq dimH νq = . χνq

We say that has Weak Almost Unique Coding (WAUC) if for all probability vectors H q with strictly positive entries there exists a set Σ (may depend on q) for BH ⊂ H which µq( ) = 0 and for every ıˆ Σ :#(ξ(ıˆ) ) = 1. BH ∈ H \BH \BH Proposition B.0.1. For any self-similar IFS on the line the conditions NDD and WUAC are equivalent.

Let δıˆ denote the Dirac-delta measure concentrated on the point ıˆ Σ . We show ∈ H the assertion in two steps. Namely, we prove that

NDD µξ(ıˆ) = δıˆ for µq-a.e. ıˆ Σ WAUC. (B.0.2) ⇐⇒ ∈ H ⇐⇒ Proof of ﬁrst equivalence in (B.0.2) . The result of Bárány–Käenmäki [BK17, Theorem 2.3.] for dimension d = 1 states that for every Bernoulli measure µq its push forward νq is exact dimensional. Moreover,

hµ H dim ν = q − , where H = log µ ([ıˆ ])dµ (ıˆ) 0. H q χ ξ(ıˆ) 1 q νq − Z ≥ From the deﬁnition of NDD we get that

NDD H = 0 µξ(ıˆ)([ıˆ1]) = 1 for µq-a.e. ıˆ Σ . ⇐⇒ ⇐⇒ ∈ H Thus it sufﬁces to show that

µξ(ıˆ)([ıˆ1]) = 1 for µq-a.e. ıˆ Σ µξ(ıˆ) = δıˆ for µq-a.e. ıˆ Σ . (B.0.3) ∈ H ⇐⇒ ∈ H The = direction in (B.0.3) is obvious. In the other direction we show that for µ -a.e. ⇐ q ıˆ Σ ∈ H µ ([ıˆ ]) = 1 = µ ([ıˆ ... ıˆ ]) = 1 for every n = µ = δ . ξ(ıˆ) 1 ⇒ ξ(ıˆ) 1 n ⇒ ξ(ıˆ) ıˆ (n) n To see the first implication fix n. Let := hıˆ1...ıˆn : (ıˆ1 ... ıˆn) 1, . . . , M (n) H { ∈ { } } and Σ be the symbolic space of infinite sequences of n-tuples (ıˆ1 ... ıˆn). There is a H ( ) natural one-to-one bijection between the elements of Σ and Σ n . A Bernoulli mea- (nH) (nH) (n) sure µq on Σ naturally defines a Bernoulli measure µq on Σ by µq ([ıˆ1 ... ıˆn]) = n H H ∏j=1 µq([ıˆj]). Applying [BK17, Theorem 2.3.] to this system yields the first implication. The second implication follows from the Monotone Convergence Theorem

1 = lim µξ(ıˆ)([ıˆ1 ... ıˆn]) = µξ(ıˆ)(ıˆ) = µξ(ıˆ) = δıˆ for µq-a.e. ıˆ Σ . n ∞ H → ⇒ ∈

Proof of second equivalence in (B.0.2). Appendix B. No Dimension Drop is equivalent to Weak Almost Unique Coding 117

= direction We claim that the conditions in the deﬁnition of WAUC are satisﬁed ⇒ with := (Σ Σ ) ıˆ Σ : µξ(ıˆ) = δıˆ . BH H \ H ∈ H 6 [ n o By assumption µq( ) = 0. Moreover,b for any ıˆ Σ we have ıˆ Σ , BH ∈ H \BH ∈ H so the probability measure µξ(ıˆ) exists and µξ(ıˆ) = δıˆ. If ˆ ξ(ıˆ) then ∈ \BH ξ(ˆ) = ξ(ıˆ), thus b δıˆ = µξ(ıˆ) = µξ(ˆ) = δˆ. That is ıˆ = ˆ. We showed that for every ıˆ Σ : ξ(ıˆ) = ıˆ . ∈ H \BH \BH { } = direction Clearly, WAUC is equivalent to the existence of with µq( ) = 0 ⇐ BH BH such that if ıˆ then ξ(ıˆ) = ıˆ . (B.0.4) 6∈ BH \BH { } Using (B.0.1) for we obtain that the set BH

Σ := ıˆ Σ : µξ(ıˆ)( ) = 0 H ∈ H BH n o has full measure: b b µq(Σ ) = 1, (B.0.5) H where we remind the reader that Σ is the set of those ıˆ Σ for which the Hb ∈ H conditional probability measure µξ(ıˆ) exists. Assume that ıˆ Σ . Then b ∈ H µξ(ıˆ) (ξ(ıˆ) ) = 1 and by (B.0.4): ξ(ıˆ) = bıˆ . \BH \BH { }

That is µξ(ıˆ)( ıˆ ) = 1 whenever ıˆ Σ . Combining this with (B.0.5) we get that { } ∈ H for a µq-full measure set of ıˆ we have µξ(ıˆ) = δıˆ. b

119

Bibliography

[AM08] Marie Albenque and Jean-Francois Marckert. “Some families of increasing planar maps”. In: Electron. J. Probab. 13 (2008), no. 56, 1624–1671. [AN04] K.B. Athreya and P.E. Ney. Branching Processes. Dover Publications, 2004. [And+05] José S. Andrade et al. “Apollonian Networks: Simultaneously Scale-Free, Small World, Euclidean, Space Filling, and with Matching Graphs”. In: Phys. Rev. Lett. 94 (2005), p. 018702. [AS04] Jamil Aouidi and Mourad Ben Slimane. “Multi-fractal formalism for non- self-similar functions”. In: Integral Transforms Spec. Funct. 15.3 (2004), pp. 189– 207. [ATK03] V. V. Aseev, A. V. Tetenov, and A. S. Kravchenko. “On Selfsimilar Jor- dan Curves on the Plane”. In: Siberian Mathematical Journal 44.3 (2003), pp. 379–386. [B15]´ Balázs Bárány. “On the Ledrappier–Young formula for self-affine measures”. In: Mathematical Proceedings of the Cambridge Philosophical Society 159.3 (2015), pp. 405–432. [BA99] A.-L. Barabási and R. Albert. “Emergence of scaling in random networks”. In: Science 286.5439 (1999), pp. 509–512. [Bar07] Krzysztof Barański.“Hausdorff dimension of the limit sets of some planar geometric constructions”. In: Advances in Mathematics 210.1 (2007), pp. 215 –245. [Bar08] Krzysztof Barański.“Hausdorff dimension of self-affine limit sets with an invariant direction”. In: Discrete Continuous Dynamical Systems - A 21.4 (2008), pp. 1015–1023. [Bar86] Michael F. Barnsley. “Fractal functions and interpolation”. In: Construc- tive Approximation 2.1 (1986), pp. 303–329. [BD06] Nicolas Broutin and Luc Devroye. “Large Deviations for the Weighted Height of an Extended Class of Trees”. In: Algorithmica 46 (2006), pp. 271– 297. [Bed84] Tim Bedford. “Crinkly curves, Markov partitions and box dimensions in self-similar sets”. PhD thesis. University of Warwick, 1984. [BG09] Jairo Bochi and Nicolas Gourmelon. “Some characterizations of domina- tion”. In: Mathematische Zeitschrift 263.1 (2009), pp. 221–231. [BHH10] S. Bhamidi, R. van der Hofstad, and G. Hooghiemstra. “First passage percolation on random graphs with finite mean degrees”. In: Ann. Appl. Probab. 20(5) (2010), pp. 1907–1965. [BHH11] S. Bhamidi, R. van der Hofstad, and G. Hooghiemstra. “First passage percolation on the Erd˝os-Rényirandom graph”. In: Combinatorics, Probability and Computing 20 (2011), pp. 683–707. 120 Bibliography

[BHH12a] S. Bhamidi, R. van der Hofstad, and G. Hooghiemstra. “Universality for first passage percolation on sparse random graphs”. arXiv:1210.6839 [math.PR]. 2012. [BHH12b] S. Bhamidi, R. van der Hofstad, and G. Hooghiemstra. “Weak disorder asymptotics in the stochastic mean-field model of distance”. In: Ann. Appl. Probab. 22(1) (2012), pp. 29–69. [BHR17] B. Bárány, M. Hochman, and A. Rapaport. “Hausdorff dimension of planar self-affine sets and measures”. In: ArXiv e-prints (Dec. 2017). arXiv: 1712.07353 [math.MG]. [BJR07] B. Bollobás, S. Janson, and O. Riordan. “The phase transition in inhomogeneous random graphs”. In: Random Structures and Algorithms 31 (2007), pp. 3–122. [BK11] Christoph Bandt and Aleksey Kravchenko. “Differentiability of fractal curves”. In: Nonlinearity 24.10 (2011), p. 2717. [BK17] Balázs Bárány and Antti Käenmäki. “Ledrappier–Young formula and exact dimensionality of self-affine measures”. In: Advances in Mathematics 318 (2017), pp. 88 –129. [BKK18] Balázs Bárány, Gergely Kiss, and István Kolossváry. “Pointwise regularity of parameterized affine zipper fractal curves”. In: Nonlinearity 31.5 (2018), p. 1705. [Bol+01] B. Bollobás et al. “The degree sequence of a scalefree random graph process”. In: Random Structures and Algorithms 18 (3 2001), pp. 279–290. [Bow08] Rufus Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Vol. 470. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008. [Boy82] D.W. Boyd. “The Sequence of Radii of the Apollonian Packing”. In: Math- ematics of Computation 19 (1982), pp. 249 –254. [BR04] B. Bollobás and O. Riordan. “The diameter of a scale-free random graph”. In: Combinatorica 24 (2004), pp. 5–34. [BRS] Balázs Bárány, MichaŁ Rams, and Károly Simon. “On the dimension of triangular self-affine sets”. In: Ergodic Theory and Dynamical Systems (). to appear, pp. 1–33. [BRS16] Balázs Bárány, Michał Rams, and Károly Simon. “On the dimension of self-affine sets and measures with overlaps”. In: Proceedings of the Ameri- can Mathematical Society 144.10 (2016), pp. 4427–4440. [BS01] Mourad Ben Slimane. “Multifractal formalism for self-similar functions expanded in singular basis”. In: Appl. Comput. Harmon. Anal. 11.3 (2001), pp. 387–419. [BS03] Mourad Ben Slimane. “Some functional equations revisited: the multifractal properties”. In: Integral Transforms Spec. Funct. 14.4 (2003), pp. 333– 348. [BS05] Julien Barral and Stéphane Seuret. “From multifractal measures to multifractal wavelet series”. In: J. Fourier Anal. Appl. 11.5 (2005), pp. 589–614. [BS12] M. Ben Sliman. “The thermodynamic formalism for the de Rham function: the increment method”. In: Izv. Ross. Akad. Nauk Ser. Mat. 76.3 (2012), pp. 3–18. Bibliography 121

[Büh71] W.J. Bühler. “Generations and degree of relationship in supercritical Markov branching processes”. In: Probability Theory and Related Fields 18(2) (1971), pp. 141–152. [CFU14] C. Cooper, A. Frieze, and R. Uehara. “The height of random k-trees and related branching processes”. In: Random Struct. Alg. 45 (2014), pp. 675– 702. [CG01] V. Choi and M. J. Golin. “Lopsided Trees, I: Analyses”. In: Algorithmica 31 (2001), pp. 240–290. [CMT14] Claire Coiffard, Clothilde Melot, and Willer Thomas. “A family of functions with two different spectra of singularities”. In: J. Fourier Anal. Appl. 20.5 (2014), pp. 961–984, [DM05] J.P.K Doye and C.P Massen. “Self-similar disk packings as model spatial scale-free networks”. In: Phys. Rev. E 71 (2005), p. 016128. [DS07] A. Darrasse and M. Soria. “Degree distribution of RAN structures and Boltzmann generation fo trees”. In: Conference on Analysis of Algorithms, AofA 07, DMTCS Proceedings. 2007, pp. 1–14. [DS17] Tushar Das and David Simmons. “The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result”. In: Inventiones mathematicae 210.1 (2017), pp. 85–134. [DZ10] A Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Vol. 38. Stochastic Modelling and Applied Probability. Springer-Verlag Berlin Heidelberg, 2010. [Ebr+13] E. Ebrahimzadeh et al. “On the Longest Paths and the Diameter in Ran- dom Apollonian Networks”. In: Electronic Notes in Discrete Mathematics 43 (2013), pp. 355–365. [ER60] Pál Erd˝osand Alfréd Rényi. “On the evolution of random graphs”. In: Publ. Math. Inst. Hungary. Acad. Sci. 5 (1960), pp. 17–61. [Fal85] K. J. Falconer. The Geometry of Fractal Sets. Cambridge Tracts in Mathe- matics. Cambridge University Press, 1985. [Fal86] K. J. Falconer. The geometry of fractal sets. Vol. 85. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1986, pp. xiv+162. [Fal88a] K. J. Falconer. “The Hausdorff dimension of self-affine fractals”. In: Math- ematical Proceedings of the Cambridge Philosophical Society 103.2 (1988), pp. 339– 350. [Fal88b] K. J. Falconer. “The Hausdorff dimension of self-affine fractals”. In: Math. Proc. Cambridge Philos. Soc. 103.2 (1988), pp. 339–350. [Fal90] K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications. Wiley, 1990. [Fal97] K. J. Falconer. Techniques in Fractal Geometry. England: John Wiley, 1997. [Fel68] W. Feller. An Introduction to Probability Theory and Its Applications. Vol. 1. Wiley, 1968. [Fen03] De-Jun Feng. “Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices”. In: Israel Journal of Mathematics 138.1 (2003), pp. 353–376. 122 Bibliography

[FH09] De-Jun Feng and Huyi Hu. “Dimension theory of iterated function systems”. In: Communications on Pure and Applied Mathematics 62.11 (2009), pp. 1435–1500. [FK18] De-Jun Feng and Antti Käenmäki. “Self-affine sets in analytic curves and algebraic surfaces”. In: Ann. Acad. Sci. Fenn. Math. 43.1 (2018), pp. 109– 119. [FL02] De-Jun Feng and Ka-Sing Lau. “The pressure function for products of non-negative matrices”. In: Math. Research Letter 9 (2002), pp. 363–378. [FM07] Kenneth Falconer and Jun Miao. “Dimensions of self-affine fractals and multifractals generated by upper-triangular matrices”. In: Fractals 15.03 (2007), pp. 289–299. [FP16] István Fazekas and Bettina Porvázsnyik. “Scale-free property for degrees and weights in an N-interactions random graph model”. In: J Math Sci 214 (2016), pp. 69–82. [Fra12] Jonathan M. Fraser. “On the packing dimension of box-like self-affine sets in the plane”. In: Nonlinearity 25.7 (2012), pp. 2075–2092. [FS14] De-Jun Feng and Pablo Shmerkin. “Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles”. In: P. Geom. Funct. Anal. 24.4 (2014), pp. 1101–1128. [FS16] Jonathan M. Fraser and Pablo Shmerkin. “On the dimensions of a family of overlapping self-affine carpets”. In: Ergodic Theory and Dynamical Systems 36.8 (2016), pp. 2463–2481. [FT12] Alan Frieze and Charalampos E. Tsourakakis. “On certain properties of random Apollonian networks”. In: Proceedings of the 9th international conference on Algorithms and Models for the Web Graph. WAW’12. Springer- Verlag, 2012, pp. 93–112. [FW05] De-Jun Feng and Yang Wang. “A Class of Self-Affine Sets and Self-Affine Measures”. In: Journal of Fourier Analysis and Applications 11.1 (2005), pp. 107– 124. [GL92] D. Gatzouras and S. P. Lalley. “Hausdorff and box dimensions of certain self-affine fractals”. In: Indiana University Mathematics Journal 41.2 (1992), pp. 533–568. [Gra+03] Ronald L. Graham et al. “Apollonian circle packings: number theory”. In: Journal of Number Theory 100.1 (2003), pp. 1 –45. [Heu98] Yanick Heurteaux. “Estimations de la dimension inférieure et de la dimension supérieure des mesures”. In: Ann. Inst. H. Poincaré Probab. Statist. 34.3 (1998), pp. 309–338. [Hoc14] Michael Hochman. “On self-similar sets with overlaps and inverse theorems for entropy”. In: Ann. of Math. 180.2 (2014), pp. 773–822. [Hu98] Huyi Hu. “Box Dimensions and Topological Pressure for some Expand- ing Maps”. In: Comm Math Phys 191.2 (1998), pp. 397–407. [Hut81] John E. Hutchinson. “Fractals and self-similarity”. In: Indiana Univ. Math. J. 30.5 (1981), pp. 713–747. [Jaf97a] S. Jaffard. “Multifractal formalism for functions. I. Results valid for all functions”. In: SIAM J. Math. Anal. 28.4 (1997), pp. 944–970. Bibliography 123

[Jaf97b] S. Jaffard. “Multifractal formalism for functions. II. Self-similar functions”. In: SIAM J. Math. Anal. 28.4 (1997), pp. 971–998. [Jan99] S. Janson. “One, two and three times log n/n for paths in a complete graph with random weights”. In: Combinatorics, Probability and Computing 8(4) (1999), pp. 347–361. [JPS07] Thomas Jordan, Mark Pollicott, and Károly Simon. “Hausdorff dimension for randomly perturbed self affine attractors”. In: Comm. Math. Phys. 270.2 (2007), pp. 519–544. [JS17] Johannes Jaerisch and Hiroki Sumi. “Pointwise Hölder exponents of the complex analogues of the Takagi function in random complex dynamics"”. In: Advances in Mathematics 313.Supplement C (2017), pp. 839 –874. [K04]¨ Antti Käenmäki. “On natural invariant measures on generalised iterated function systems”. In: Ann. Acad. Sci. Fenn. Math. 29.2 (2004), pp. 419–458. [KK15] I. Kolossváry and Júlia Komjáthy. “First Passage Percolation on Inhomo- geneous Random Graphs”. In: Advances in Applied Probability 47.2 (2015), pp. 1–22. [KKV16] István Kolossváry, Júlia Komjáthy, and Lajos Vágó. “Degrees and distances in random and evolving apollonian networks”. In: Advances in Applied Probability 48.3 (2016), 865–902. [KR89] Sanjiv Kapoor and Edward M. Reingold. “Optimum lopsided binary trees”. In: J. ACM 36 (1989), pp. 573–590. [KS18] I. Kolossváry and K. Simon. “Triangular Gatzouras-Lalley-type planar carpets with overlaps”. In: ArXiv e-prints (July 2018). arXiv: 1807.09717 [math.MG]. [Lal88] Steven P. Lalley. “The Packing and Covering Functions of Some Self- similar Fractals”. In: Indiana University Mathematics Journal 37.3 (1988), pp. 699–709. [Led92] François Ledrappier. “On the dimension of some graphs”. In: Symbolic dynamics and its applications (New Haven, CT, 1991). Vol. 135. Contemp. Math. Amer. Math. Soc., Providence, RI, 1992, pp. 285–293. [LY85a] F. Ledrappier and L.-S. Young. “The Metric Entropy of Diffeomorphisms: Part I: Characterization of Measures Satisfying Pesin’s Entropy Formula”. In: Annals of Mathematics 122.3 (1985), pp. 509–539. [LY85b] F. Ledrappier and L.-S. Young. “The Metric Entropy of Diffeomorphisms: Part II: Relations between Entropy, Exponents and Dimension”. In: An- nals of Mathematics 122.3 (1985), pp. 540–574. [Mat95] Pertti Mattila. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics. Cam- bridge University Press, 1995. [McM84] Curt McMullen. “The Hausdorff dimension of general Sierpińskicar- pets”. In: Nagoya Mathematical Journal 96 (1984), pp. 1–9. [Mor16] Ian D. Morris. “An inequality for the matrix pressure function and applications”. In: Advances in Mathematics 302 (2016), pp. 280–308. [Mor17] Ian D. Morris. “On Falconer’s formula for the generalised Rényi dimension of a self-affine measure”. In: Ann. Acad. Sci. Fenn. Math. 42.1 (2017), pp. 227–238. 124 Bibliography

[Nik04] P.P. Nikitin. “The Hausdorff Dimension of the Harmonic Measure on de Rham’s Curve”. In: Journal of Mathematical Sciences 121.3 (2004), pp. 2409– 2418. [Oka16] Kazuki Okamura. “On regularity for de Rham’s functional equations”. In: Aequationes mathematicae 90.6 (2016), pp. 1071–1085. [Ose68] V.I. Oseledets. “A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems”. In: Trans. Moscow Math. Soc. 19 (1968), pp. 197–231. [Per94] Yuval Peres. “The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure”. In: Mathematical Proceedings of the Cambridge Philosophical Society 116.3 (1994), 513–526. [PG15] Vladimir Yu. Protasov and Nicola Guglielmi. “Matrix approach to the global and local regularity of wavelets”. In: Poincare J. Anal. Appl. 2 (2015), pp. 77–92. [Pro04] V Yu Protasov. “On the regularity of de Rham curves”. In: Izvestiya: Math- ematics 68.3 (2004), p. 567. [Pro06] V Yu Protasov. “Fractal curves and wavelets”. In: Izvestiya: Mathematics 70.5 (2006), p. 975. [PS] Leticia Pardo-Simón. “Dimensions of an overlapping generalization of Barańskicarpets”. In: Ergodic Theory and Dynamical Systems (), pp. 1–31. [PS00] Yuval Peres and Wilhelm Schlag. “Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions”. In: Duke Math. J. 102.2 (Apr. 2000), pp. 193–251. [PS95] M. Pollicott and K. Simon. “The Hausdorff dimension of λ-expansions with deleted digits”. In: Trans. Amer. Math. Soc. 347 (1995), pp. 967–983. [PS96] Y. Peres and B. Solomyak. “Absolute continuity of Bernoulli convolutions, a simple proof”. In: Math. Research Letters 3.3 (1996), pp. 231–239. [PU89] F. Przytycki and M. Urbanski. “On the Hausdorff dimension of some fractal sets”. In: Studia Mathematica 93.2 (1989), pp. 155–186. [PV15] Mark Pollicott and Polina Vytnova. “Estimating singularity dimension”. In: Math. Proc. Camb. Phil. Soc. 158.02 (2015), pp. 223–238. [Rha47] Georges de Rham. “Une peu de mathématiques à propos d’une courbe plane”. In: Elemente der Math. 2.4 (1947), 73–76, 89––97. [Rha56] Georges de Rham. “Sur une courbe plane”. In: J. Math. Pures Appl. 35 (1956), pp. 25–42. [Rha59] Georges de Rham. “Sur les courbes limités de polygones obtenus par trisection”. In: Enseignement Math. 5 (1959), pp. 29–43. [RS03] M. Rams and K. Simon. “Hausdorff and packing measure for solenoids”. In: Ergodic Theory and Dynamical Systems 23.1 (2003), pp. 273–291. [Sak05] Koichi Saka. “Scaling exponents of self-similar functions and wavelet analysis”. In: Proc. Amer. Math. Soc. 133.4 (2005), pp. 1035–1045. [Seu16] Stéphane Seuret. “Multifractal Analysis and Wavelets”. In: New Trends in Applied Harmonic Analysis: Sparse Representations, Compressed Sensing, and Multifractal Analysis. Ed. by Akram Aldroubi et al. Cham: Springer International Publishing, 2016, pp. 19–65. Bibliography 125

[Sim12] David Simmons. “Conditional measures and conditional expectation; Rohlin’s Disintegration Theorem”. In: Discrete & Continuous Dynamical Systems-A 32.7 (2012), pp. 2565–2582. [Sol95] Boris Solomyak. “On the Random Series ∑ λn (an Erdos Problem)”. In: ± Annals of Mathematics 142.3 (1995), pp. 611–625. [Sol98a] Boris Solomyak. “Measure and dimension for some fractal families”. In: Mathematical Proceedings of the Cambridge Philosophical Society 124.3 (1998), pp. 531–546. [Sol98b] Boris Solomyak. “Measure and dimension for some fractal families”. In: Math. Proc. Cambridge Philos. Soc. 124.3 (1998), pp. 531–546. [Zha+06] Zhongzhi Zhang et al. “High-dimensional Apollonian networks”. In: Jour- nal of Physics A: Mathematical and General 39.8 (2006), p. 1811. [Zha+08] Zhongzhi Zhang et al. “Analytical solution of average path length for Apollonian networks”. In: Phys. Rev. E 77 (2008), p. 017102. [ZRC06] Zhongzhi Zhang, Lili Rong, and Francesc Comellas. “High-dimensional random Apollonian networks”. In: Physica A: Statistical Mechanics and its Applications 364 (2006), pp. 610 –618. [ZRZ06] Zhongzhi Zhang, Lili Rong, and Shuigeng Zhou. “Evolving Apollonian networks with small-world scale-free topologies”. In: Phys. Rev. E 74 (4 2006), p. 046105. [ZYW05] Tao Zhou, Gang Yan, and Bing-Hong Wang. “Maximal planar networks with large clustering coefficient and power-law degree distribution”. In: Phys. Rev. E 71 (2005), p. 046141. [Fen19] De-Jun Feng. “Dimension of invariant measures for affine iterated function systems”. In: arXiv e-prints, arXiv:1901.01691 (Jan. 2019), arXiv:1901.01691. arXiv: 1901.01691. [Hoc15] M. Hochman. “On self-similar sets with overlaps and inverse theorems for entropy in Rd”. In: ArXiv e-prints (Mar. 2015). arXiv: 1503.09043. [Mor18] Ian D. Morris. “Fast approximation of the affinity dimension for dominated affine iterated function systems”. In: arXiv e-prints, arXiv:1807.09084 (July 2018), arXiv:1807.09084. arXiv: 1807.09084. [Ota17] A. Otani. “Fractal dimensions of graph of Weierstrass-type function and local Hölder exponent spectra”. In: Nonlinearity 31.1 (2017), pp. 263–292.