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Numerical Values of the Hausdorff and Packing Measures for Limit NUMERICAL VALUES OF THE HAUSDORFF AND PACKING MEASURES FOR LIMIT SETS OF ITERATED FUNCTION SYSTEMS James Edward Reid Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS August 2017 APPROVED: Mariusz Urbański, Major Professor Lior Fishman, Committee Member Stephen Jackson, Committee Member Charles Conley, Chair of the Department of Mathematics David Holdeman, Dean of the College of Arts and Science Victor R. Prybutok, Dean of the Toulouse Graduate School Reid, James Edward. Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems. Doctor of Philosophy (Mathematics), August 2017, 107 pp., 56 numbered references. In the context of fractal geometry, the natural extension of volume in Euclidean space is given by Hausdorff and packing measures. These measures arise naturally in the context of iterated function systems (IFS). For example, if the IFS is finite and conformal, then the Hausdorff and packing dimensions of the limit sets agree and the corresponding Hausdorff and packing measures are positive and finite. Moreover, the map which takes the IFS to its dimension is continuous. Developing on previous work, we show that the map which takes a finite conformal IFS to the numerical value of its packing measure is continuous. In the context of self-similar sets, we introduce the super separation condition. We then combine this condition with known density theorems to get a better handle on finding balls of maximum density. This allows us to extend the work of others and give exact formulas for the numerical value of packing measure for classes of Cantor sets, Sierpinski N-gons, and Sierpinski simplexes. Copyright 2017 by James Edward Reid ii ACKNOWLEDGMENTS A special thanks to my advisor Professor Urba´nski.Thank you! And of course, thank you to my family for their unconditional support: Mom, Dad, Ryan, Geemaw, Pa, Granny, Poppa, Matt, Richard, Sherri, Victoria, Katherine, Mark, Teresa, and Cooper. Thank you! iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii CHAPTER 1 INTRODUCTION1 1.1. Hausdorff and Packing Measures1 1.2. Iterated Function Systems4 1.3. Conformal Mappings5 1.4. Conformal Iterated Function Systems9 CHAPTER 2 CONTINUITY OF INVARIANT SETS, INVARIANT MEASURES, AND HAUSDORFF/PACKING DIMENSION 13 2.1. The Hausdorff Metric and Continuity of Limit Sets 13 2.2. The Space of Measures and Metrics in the Weak*-Topology 19 2.3. The Transfer Operator and Continuity of Invariant Measures 22 2.4. The Pressure Function and Continuity of Hausdorff/Packing Dimension 36 CHAPTER 3 CONTINUITY OF THE NUMERICAL VALUE OF HAUSDORFF AND PACKING MEASURES FOR FINITE ITERATED FUNCTION SYSTEMS 42 3.1. General Density Theorems 43 3.2. Density Theorems for Self-Similar Sets 45 3.3. Density Theorems for Self-Conformal Sets 48 3.4. Continuity of the Numerical Value for Conformal IFS 67 CHAPTER 4 THE NUMERICAL VALUE OF PACKING MEASURE FOR SUPER SEPARATED ITERATED FUNCTION SYSTEMS 80 4.1. The Super Separation Condition 81 iv 4.2. A Density Theorem for Super Separated IFS 83 4.3. Packing Measure of Super Separated Cantor Sets 85 4.4. Packing Measure of Super Separated Sierpi´nskiN-gons 91 4.5. Packing Measure of Super Separated Sierpi´nskiSimplexes 102 BIBLIOGRAPHY 104 v CHAPTER 1 INTRODUCTION 1.1. Hausdorff and Packing Measures We first discuss Hausdorff and packing measures, which are fundamental to geometric measure theory and fractal geometry (see for example [17, 18, 31, 35]). In this section we will assume that (X; ρ) is a metric space. Definition 1.1.1. The diameter of a nonempty set A ⊆ X is defined to be jAj = supfρ(x; y): x; y 2 Ag; and we define j;j = 0. Definition 1.1.2. Suppose 0 < δ ≤ 1.A δ-covering of a set A ⊆ X is a countable [ collection U of subsets of X with diameters at most δ such that A ⊆ U. U2U Definition 1.1.3. Let s ≥ 0 be some real number. For every 0 < δ ≤ 1, the s-dimensional δ-Hausdorff premeasure of a set A ⊆ X is given by ( ) s X s Hδ (A) = inf jUj : U is a δ-covering of A : U2U The s-dimensional Hausdorff measure of a set A ⊆ X is given by s s H (A) = sup Hδ (A) : δ>0 The Hausdorff dimension of A is defined to be HD (A) = inf fs ≥ 0 : Hs (A) = 0g = sup fs ≥ 0 : Hs (A) = 1g [ f0g: The integer-dimensional Hausdorff measures are important cases when X = Rd be- cause they correspond with our usual notion of length, area, and volume (see [18, 31] for more precise statements): (i) H0 is the counting measure, 1 (ii) H1 (Γ) is the classical notion of length for a rectifiable curve Γ, (iii) if E is a k-dimensional C1-surface with 1 ≤ k ≤ d − 1, Hk (E) is a constant multiple of the classical k-dimensional area of E, (iv) Hd is a constant multiple of the d-dimensional Lebesgue measure. In addition, for any s ≥ 0, if 0 < Hs (A) < 1, then s = HD (A). Therefore, nice sets such as k-dimensional C1-surfaces retain their usual dimension. However, Hausdorff dimension allows for non-integer dimensions, and many interesting fractal sets have non- integer Hausdorff dimension and can even have positive and finite Hausdorff measure. For these reasons, we say that Hausdorff measure extends the notion of volume in Rd, and Hausdorff dimension extends the usual notion of dimension. In fractal geometry, Hausdorff dimension can be considered an invariant because it is preserved under bi-Lipschitz maps. Another extension of volume in Rd called packing measure was introduced by Sullivan [47] and Taylor and Tricot [49, 50, 51]. Packing measures exhibit many of the same properties as Hausdorff measures. However, Sullivan demonstrated that for geometric limit sets arising from Kleinian groups, sometimes Hausdorff measure is the natural measure to consider, and sometimes packing measure is the natural measure to consider. More specifically, he showed that sometimes the Hausdorff measure of the limit set is 0, but the packing measure is positive and finite. Taylor and Tricot were interested in Brownian trajectories and showed that the packing measure with a certain gauge function is positive and finite. Packing measure is a sort of dual to Hausdorff measure in the sense that Hausdorff measure is defined in terms of efficient coverings, while packing measure is defined in terms of bountiful packings. Definition 1.1.4. Suppose 0 < δ < 1.A δ-packing of a set A ⊆ X is a countable collection 1 of closed balls fB(xi; ri)gi=1 with centers in A, radii at most δ, and ρ(xi; xj) > ri + rj for all i 6= j. Definition 1.1.5. Let s ≥ 0 be some real number. For every 0 < δ ≤ 1, the s-dimensional 2 δ-packing premeasure of a set A ⊆ X is given by ( ) s X s Pδ (A) = sup jBj : B is a δ-packing of A : B2B The s-dimensional packing premeasure of a set A ⊆ X is given by s s P0 (A) = inf Pδ (A) : δ>0 The s-dimensional packing measure of a set A ⊆ X is given by ( 1 1 ) s X s [ P (A) = inf P0 (Ai): A ⊆ Ai : i=1 i=1 The packing dimension of A is defined to be PD (A) = inf fs ≥ 0 : Ps (A) = 0g = sup fs ≥ 0 : Ps (A) = 1g [ f0g: For any set A ⊆ Rd and s ≥ 0, Hs (A) ≤ Ps (A) (Theorem 5.12 [35]). However, it is actually rare for equality to hold. It turns out that 0 < Hs (A) = Ps (A) < 1 if and only if s is an integer and A is sufficiently rectifiable (see Theorem 17.11 [35]). In particular, we see that packing measure is an extension of volume just like Hausdorff measure. However, the numerical value of their measures may behave quite differently for fractal sets. We will be especially interested in the exact numerical value of packing measure. However, the definition is given in three stages, and this can make it awkward to work with at times. Fortunately, we have the following result by Feng, Hua, and Wen [20], which will be of great use in this thesis. Theorem 1.1.1. Let K be a compact subset of Rd, and let s ≥ 0. s s s (1) If P0 (K) < 1, then P (K) = P0 (K). s (2) If P0 (K) = 1, then for any " > 0 there exists a compact subset F of K such that s s s s P (F ) = P0 (F ) and P (F ) ≥ P (K) − ". 3 1.2. Iterated Function Systems Let (X; ρ) be a nonempty compact metric space. Fix a countable (finite or infinite) set I with at least two elements. We will refer to the set I as the alphabet. An iterated function system (IFS) is a collection Φ = f'igi2I , where 'i : X ! X is a contraction for all i 2 I and supi2I Lip ('i) < 1. We will almost always use a capital letter to denote an IFS and the corresponding lowercase letter for the mappings in the IFS. The collection of all iterated function systems on the space X with alphabet I is denoted IFS (X; I). A finite word of length n ≥ 1 in the alphabet I is a finite sequence of symbols (a tuple) denoted by ! = !1 :::!n. The length of the word ! is denoted by j!j = n. For each n ≥ 1, we define In to be the set of all finite words in the alphabet I of length n. Define ∗ S1 n ∗ I = n=1 I be the collection of all finite words.
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