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Fractal, Group Theoretic, and Relational Structures on Cantor Space

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Fractal, Group Theoretic, and Relational Structures on Cantor Space View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository FRACTAL, GROUP THEORETIC, AND RELATIONAL STRUCTURES ON CANTOR SPACE Casey Ryall Donoven A Thesis Submitted for the Degree of PhD at the University of St Andrews 2016 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/11370 This item is protected by original copyright Fractal, Group Theoretic, and Relational Structures on Cantor Space Casey Ryall Donoven This thesis is submitted in partial fulfilment for the degree of PhD at the University of St Andrews November 2, 2016 Abstract Cantor space, the set of infinite words over a finite alphabet, is a type of metric space with a `self-similar' structure. This thesis explores three areas concerning Cantor space with regard to fractal geometry, group theory, and topology. We first find results on the dimension of intersections of fractal sets within the Cantor space. More specifically, we examine the intersection of a subset E of the n-ary Cantor space, Cn with the image of another subset F under a random isometry. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. We then consider a class of groups, denoted by Vn(G), of homeomorphisms of the Cantor space built from transducers. These groups can be seen as homeomorphisms that respect the self-similar and symmetric stucture of Cn, and are supergroups of the Higman-Thompson groups Vn. We explore their isomorphism classes with our primary result being that Vn(G) is isomorphic to (and conjugate to) Vn if and only if G is a semiregular subgroup of the symmetric group on n points. Lastly, we explore invariant relations on Cantor space, which have quotients homeo- morphic to fractals in many different classes. We generalize a method of describing these quotients by invariant relations as an inverse limit, before characterizing a specific class of fractals known as Sierpi´nskirelatives as invariant factors. We then compare relations arising through edge replacement systems to invariant relations, detailing the conditions under which they are the same. i Declarations I, Casey Donoven, hereby certify that this thesis, which is approximately 32,000 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in September 2012 and as a candidate for the degree of Ph.D in September 2013; the higher study for which this is a record was carried out in the University of St Andrews between 2012 and 2016. Date: ....................... Signature of candidate: ............................... I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of Ph.D in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date: ....................... Signature of supervisor: ............................... Date: ....................... Signature of supervisor: ............................... ii In submitting this thesis to the University of St Andrews I understand that I am giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. I also understand that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that my thesis will be electronically accessible for personal or research use unless exempt by award of an embargo as requested below, and that the library has the right to migrate my thesis into new electronic forms as required to ensure continued access to the thesis. I have obtained any third-party copyright permissions that may be required in order to allow such access and migration, or have requested the appropriate embargo below. The following is an agreed request by candidate and supervisor regarding the publication of this thesis: PRINTED COPY No embargo on print copy ELECTRONIC COPY No embargo on electronic copy Date: ....................... Signature of candidate: ............................... Signature of supervisor: ............................... Signature of supervisor: ............................... iii Acknowledgements Thank you to all of the people who made this possible: to Kenneth and Collin, for their dedication to my academic career; to my fellow PhD students, for all the games and crosswords; to Todd, Margaret, Brent, Bri, and my whole family, for their support from across the pond; to Ian and Teresa, for being my home away from home; to Jenny, for everything. iv Contents 1 Cantor Space: An Introduction 1 1.1 The Cantor Space Cn .............................1 1.1.1 Definitions . .1 1.1.2 Cantor Topology . .3 1.1.3 Cantor Metrics . .6 1.2 Equivalent Representations of Cantor Space . .6 1.2.1 The Infinite Tree Tn .........................7 1.2.2 Middle Thirds Cantor Set . .8 2 Codimension Formulae 12 2.1 Introduction . 12 2.1.1 Fractal Dimension . 13 2.1.2 Fractal Codimension Formulae in Rn ................ 15 2.1.3 Fractal Codimension Formulae in Cn ................ 17 2.2 Upper Box-Counting Dimension: Upper Bound . 20 2.3 Hausdorff Dimension: Upper Bound . 22 2.4 Hausdorff Dimension: Lower Bound . 26 2.4.1 Martingales . 27 2.4.2 Lower Bound of Hausdorff Dimension . 28 3 Thompson's Group V and Family 40 3.1 Homeomorphisms as Transducers . 40 v 3.1.1 Definitions and Examples . 41 3.1.2 The Rational Group . 44 3.1.3 Thompson Groups and Generalizations . 46 3.1.4 Statement of Results . 50 3.2 Proof of Theorems 3.10 and 3.11 . 52 3.2.1 Semiregular Permutation Groups . 52 3.2.2 Rubin Conjugators . 55 3.2.3 Proof of Reverse Implication in Theorem 3.10 . 59 3.2.4 Example Isomorphism Classes . 64 4 Invariant Factors 68 4.1 Invariant Relations . 69 4.1.1 Construction . 69 4.1.2 Small Relations on the Binary Cantor Space . 78 4.2 Sierpi´nskiRelatives . 85 4.2.1 Background and Definitions . 85 4.2.2 D as an Invariant Factor . 88 4.2.3 Sierpi´nskiRelatives as Invariant Factors . 95 4.3 Edge Replacement Systems . 100 4.3.1 Construction . 100 4.3.2 Characterizing Gluing Relations . 104 4.3.3 Invariant Gluing Relations . 109 5 Directions for Future Work 113 5.1 Codimension Formulae . 113 5.2 Thompson Group Generalisations . 114 5.3 Invariant Relations . 116 vi Chapter 1 Cantor Space: An Introduction Cantor space is a topological and metric space, that exhibits many self-similar properties. We explore the consequences of this regularity in fractal geometry, group theory, and topology. We begin by defining Cantor space and its topology and metric before giving several constructions of equivalent representations. These representations allow the structure of Cantor space to be displayed visually and provide alternative approaches to its study. 1.1 The Cantor Space Cn Our first description of Cantor space, and main focus for the introduction, is perhaps the least visual but most useful of the three constructions we present, giving a foundation on which to build a topological space and metric space. 1.1.1 Definitions Let n 2 N with n ≥ 2 and let Xn = f0; 1; : : : ; n − 1g. The n-ary Cantor set is then ! the set of infinite words over the alphabet Xn, denoted by Cn = Xn . This is also known the full one-sided shift space when equipped with the shift map, see [30]. To establish consistent notation, we use lower case Roman letters as individual letters in the n-ary 1 alphabet Xn and lower case Greek letters as words (finite and infinite) over the alphabet. We will usually state whether or not a word is finite but it should generally be clear from context. ∗ The set of all finite words over the alphabet Xn is denoted by Xn, which includes the ∗ empty word . The length of a word α 2 Xn, denoted by jαj, is the number of letters k that make up α (which is 0 for the empty word). We will use Xn to denote the set of words of length k. Let α be a finite word of length m > 0 and let k 2 N be such that 0 < k ≤ m. We us the notation αk to represent the kth letter of α and αjk = α1α2 : : : αk to represent the word formed by the first k letters of α.A prefix of α is any word consisting of the first l letters of α (such as αjl), for some 0 ≤ l ≤ m, and a suffix of α is a word consisting of the last l letters of α. Note that l may be 0 or m, which implies that the each finite word is its own prefix and suffix while the empty word is a prefix and suffix of every word. The definition of prefix and suffix (and the notation αjk) also extends to infinite words. We write α 4 β if α is a prefix of β and α ? β if neither α nor β are prefixes of one another. The meet of two finite or infinite words, α and β, denoted by α ^ β is the ∗ longest word γ such that γ 4 α and γ 4 β.
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