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Fractals in Dimension Theory and Complex Networks

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Fractals in Dimension Theory and Complex Networks BUDAPEST UNIVERSITY OF TECHNOLOGY AND 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. v 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 L ........................... 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 L = dimB L ................... 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 w = 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 ix List of Symbols a (A) ... a (A) singular values of a d d matrix A 1 ≥ ≥ d × a(x), ar(x) pointwise and regular Hölder exponent (3.1.3) , , , Iterated Function Systems F G H T G, L, W attractor of an Iterated Function System S, S symbolic spaces H P, P , P , p natural projections from a symbolic space s Hs H d, (d-approximate) s-dimensional Hausdorff measure (A.0.1) H H2 md, sd e expectation and variance of Yd (4.1.3) p, q, l probability vectors p∗ optimal vector for Hausdorff dimension p optimal vector for box dimension mp Bernoulli measure on a symbolic space nep push forward of mp fs singular value function (A.0.4) D(p) formula for dimH mp (2.2.1) E(b), Er(b) b-level set of a(x) and ar(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 xi 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 1 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 geomet- ric measure theory, dimension theory, dynamical systems, number theory, analysis, differential equations or probability theory to name a few.
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