BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS DOCTORAL THESIS BOOKLET 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 1 Introduction The word "fractal" comes from the Latin fractus¯ meaning "broken" or "fractured". Since the 1970s, 80s, fractal geometry has become an important area of mathematics with many connections to theory and practice alike. The main aim of the Thesis is to demonstrate the 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 dif- ferent 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. Chapter 1 of the thesis gives an introduction to these topics and informally ex- plains the contributions made in an accessible way to a wider mathematical audience. Chapters 2, 3 and 4 contain the precise definitions and rigorous formulations of our re- sults, together with the proofs. They are based on the papers [KS18; BKK18; KKV16], respectively. 1 Self-affine planar carpets A self-affine Iterated Function System (IFS) F consists of a finite collection of maps d d fi : R ! R of the form fi(x) = Aix + ti, d×d for i 2 [N] := f1, 2, . , Ng, where Ai 2 R is a contracting, invertible matrix and d ti 2 R is a translation vector. The self-affine set corresponding to F is the unique non-empty compact set L, called the attractor, which satisfies [ L = fi(L). i2[N] Perhaps the most fundamental question in fractal geometry is to determine the different notions of dimension of a set. The thesis deals with Hausdorff, packing, (lower and upper) box, and affinity dimension denoted by dimH, dimP, dimB, dimB dimB, and dimAff, respectively. All self-affine sets satisfy dimH L ≤ dimP L ≤ dimBL ≤ minfdimAff L, dg. Equality of these dimensions is typical in some generic sense [Fal88; Sol98a; BHR17]. However, in specific cases strict inequality is possible. Planar carpets have drawn much attention, because they form a large class of examples in R2 for which this ex- ceptional behavior is typical. 2 Gatzouras–Lalley-type planar carpets Gatzouras–Lalley (GL) carpets [GL92] are the attractors of self-affine IFSs on the plane 2 whose first level cylinders (i.e. the images fi([0, 1] )) are aligned into columns (of possibly different widths) using orientation preserving maps with linear parts given by diagonal matrices. Moreover, they assumed the Rectangular Open Set Condition 2 (ROSC), i.e. the interiors of fi([0, 1] ) are disjoint, and that the width of a rectangle is larger than its height, we say that direction-x dominates. Later, Bara´nski[Bar08] kept these assumptions, but allowed lower triangular ma- 2 trices instead of diagonal. As a result, the images fi([0, 1] ) are parallelograms with two vertical sides. We call them Triangular Gatzouras–Lalley-type (TGL) planar car- pets, 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 Figure1 show the images of [0, 1]2 under the maps defining a GL carpet on the left and a TGL carpet on the right. 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. Even though the ROSC holds, it is not immediate that the dimension of the two attractors should be the same. FIGURE 1: The IFS defining a Gatzouras–Lalley carpet on left and trian- gular Gatzouras–Lalley-type carpet on right, which are brothers. Other constructions for planar carpets include works of Bedford and McMullen[McM84; Bed84], Bara´nsky[Bar07], Feng–Wang [FW05] and Fraser [Fra12]. Only recently were carpets with overlaps considered, see Fraser–Shmerkin [FS16] and Pardo-Simón [PS]. The study of constructions with overlaps is always much more involved than the case when ROSC is assumed. Our goal was to study TGL carpets with overlaps. Our construction Our main contribution was to continue the works [Bar08; GL92] to allow different types of overlaps in the TGL construction. This is best illustrated with Figure2. All three examples are brothers of the GL carpet in Figure1. On the left, the columns are shifted in a way that the IFS on the x-axis generated by the columns, denoted by H, satisfies Hochman’s Exponential Separation Condition (HESC), see Definition 1.7. This type of shifted columns were considered in [FS16] and [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, see Definition 1.7. The one on the right on Figure2 has both types of overlaps. 3 FIGURE 2: Triangular Gatzouras–Lalley-type carpets with different overlaps. Left: shifted columns satisfying Hochman’s exponential sep- aration condition. Center: non-overlapping columns, transversality condition. Right: mixture of both. To fix notation, here is the formal definition. Denote the closed unit square by R = [0, 1] × [0, 1]. Let A = fA1,..., ANg be a family of 2 × 2 invertible, strictly con- tractive, real-valued lower triangular matrices. The corresponding self-affine IFS is the collection of affine maps N bi 0 ti,1 F = f fi(x) := Aix + tigi=1, where Ai = and ti = , (1.1) di ai ti,2 for translation vectors ti, with ti,1, ti,2 ≥ 0. We assume that ai, bi 2 (0, 1). Orthogonal projection of F to the horizontal x-axis, denoted projx, generates an important self-similar IFS on the line N He = fehi(x) := bix + ti,1gi=1. F H = We denote the attractor of and e by L LF and LHe respectively. Definition 1.1. We say that an IFS of the form (1.1) is Triangular Gatzouras-Lalley type and we call its attractor L a TGL planar carpet if the following conditions hold: (a) direction-x dominates, i.e. 0 < ai < bi < 1 for all i 2 [N] := f1, 2, . , Ng, (b) column structure: there exists a partition of [N] into M > 1 sets I1,..., IM with cardi- nality jIıˆj = Nıˆ > 0 so that I1 = f1, . , N1g and Iıˆ = fN1 + ... + Nıˆ−1 + 1, . , N1 + ... + Nıˆg for ıˆ = 2, . , M. Assume that for two distinct indices k and ` 2 f1, . , Ng ( bk = b` =: rıˆ, if there exists ıˆ 2 f1, . , Mg such that k, `2 Iıˆ, then (1.2) tk,1 = t`,1 =: uıˆ. We also introduce M H = fhıˆ(x) := rıˆx + uıˆgıˆ=1, H and we observe that the attractor LH of is identical with LHe. 4 ≤ 2 f g (c) we assume that ∑j2Iıˆ aj 1 holds for every ıˆ 1, . , M and the non-overlapping column structure uıˆ + rıˆ ≤ uıˆ+1 for ıˆ = 1, . , M − 1 and uM + rM ≤ 1. (1.3) (d) Without loss of generality we always assume in this paper that (A1) fi(R) ⊂ R for all i 2 [N] and (A2) The smallest and the largest fixed points of the functions of H are 0 and 1 respec- tively. Observe that the definition allows overlaps within columns (like the second figure in Figure2), but columns do not overlap. We say that L is a shifted TGL carpet if we drop the assumption (1.3), that is non- M overlapping column structure is NOT assumed, we require only that ∑ıˆ=1 rıˆ ≤ 1 (like the first figure in Figure2). Definition 1.2. We say that a shifted TGL carpet L has uniform vertical fibres if s−sH ∑ aj = 1 for every ıˆ 2 [M], j2Iıˆ where s = dimB L and sH = dimB LH. Furthermore, we call L a diagonally homogeneous shifted TGL carpet if bi ≡ b and ai ≡ a for every i 2 [N]. In particular, a diagonally homogeneous carpet has uniform vertical fibres if N/M 2 N and Nıˆ = N/M for every ıˆ 2 f1, . , Mg. Definition 1.3. A self-affine IFS Fe is a Gatzouras-Lalley IFS and its attractor Le is a GL carpet if Fe is a TGL IFS as in Definition 1.1 with the additional assumptions that all off-diagonal elements di = 0 and the rectangular open set condition (ROSC) holds. Definition 1.4. Let L be a shifted TGL carpet generated from the IFS F of the form (1.1). We say that the Gatzouras–Lalley IFS ˜ ˜ ˜ ˜ ˜ N ˜ bi 0 ˜ ti,1 Fe = f fi(x) := Aix + tigi=1, where Ai = and ti = , 0 a˜i t˜i,2 and its attractor Le is the GL brother of F and L, respectively, if a˜i = ai and b˜i = bi for every i 2 [N], furthermore, Fe has the same column structure (1.2) as F. If the shifted TGL carpet L is actually a TGL (that is L has non-overlapping column structure) then we also require that eti,1 = ti,1 holds for all i 2 [N]. M There always exists such a brother since we assume Definition 1.1 (c) and ∑ıˆ=1 rıˆ ≤ 1. Throughout, the GL brother of L will always be denoted with the extra tilde Le.