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Sara Munday Phd Thesis FINITE AND INFINITE ERGODIC THEORY FOR LINEAR AND CONFORMAL DYNAMICAL SYSTEMS Sara Ann Munday A Thesis Submitted for the Degree of PhD at the University of St. Andrews 2012 Full metadata for this item is available in Research@StAndrews:FullText at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/3220 This item is protected by original copyright Finite and Infinite Ergodic Theory for Linear and Conformal Dynamical Systems. S. Munday i Abstract The first main topic of this thesis is the thorough analysis of two families of piecewise linear maps on the unit interval, the a-Lüroth and a-Farey maps. Here, a denotes a countably infinite partition of the unit interval whose atoms only accumulate at the origin. The basic properties of these maps will be developed, including that each a-Lüroth map (denoted La ) gives rise to a series expansion of real numbers in [0;1], a certain type of Generalised Lüroth Series. The first example of such an expansion was given by Lüroth. The map La is the jump transformation of the corresponding a-Farey map Fa . The maps La and Fa share the same relationship as the classical Farey and Gauss maps which give rise to the continued fraction expansion of a real number. We also consider the topological properties of Fa and some Diophantine-type sets of numbers expressed in terms of the a-Lüroth expansion. Next we investigate certain ergodic-theoretic properties of the maps La and Fa . It will turn out that the Lebesgue measure l is invariant for every map La and that there exists a unique Lebesgue-absolutely continuous invariant measure for Fa . We will give a precise expression for the density of this measure. Our main result is that both La and Fa are exact, and thus ergodic. The interest in the invariant measure for Fa lies in the fact that under a particular condition on the underlying partition a, the invariant measure associated to the map Fa is infinite. Then we proceed to introduce and examine the sequence of a-sum-level sets arising from the a-Lüroth map, for an arbitrary given partition a. These sets can be written dynamically in terms of Fa . The main result concerning the a-sum-level sets is to establish weak and strong renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate estimates in renewal theory. Our final main result concerning the a-Lüroth and a-Farey systems is to provide a fractal- geometric description of the Lyapunov spectra associated with each of the maps La and Fa . The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the systems we consider are always piecewise linear. However, since our analysis is based on a large family of different partitions of U , the class of maps which we consider in this paper allows us to detect a variety of interesting new phenomena, including that of phase transitions. Finally, we come to the conformal systems of the title. These are the limit sets of discrete subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type sets contained in the limit set that are similar to those considered for the maps La . These sets are then used in our second main result to analyse the more geometrically defined strict-Jarník limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson measure associated to the Fuchsian group. ii iii Declaration I, Sara Munday, hereby certify that this thesis, which is approximately 25000 words in length, has been written by me, that it is the record of work carried out by me and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in September 2006 and as a candidate for the degree of Doctor of Philosophy in September 2007; the higher study for which this is a record was carried out in the University of St Andrews between 2007 and 2011. Date Signature of candidate I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regula- tions appropriate for the degree of Doctor of Philosophy 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 1 Date Signature of supervisor 2 In submitting this thesis to the University of St Andrews I understand that I am giving per- mission 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 electronic pub- lication of this thesis: Access to printed copy and electronic publication of thesis through the University of St Andrews. Date Signature of candidate Date Signature of supervisor 1 Date Signature of supervisor 2 Copyright Notice The copyright of this thesis rests with the author. No quotation from it should be published without her prior written consent and information derived from it should be acknowledged. iv v Acknowledgements I would first like to offer my sincerest thanks to my supervisor Prof Dr Bernd O. Stratmann. Without his constant patience and encouragement this thesis would never have been written. I would also like to heartily thank Dr James D. Mitchell, for agreeing to become my supervisor at the last moment and helping me to improve the presentation of my work. For numerous helpful and stimulating discussions, I would like to thank Prof Dr Marc Kesseböhmer, Dr Kurt Falk and other members of the research groups in Bremen and St Andrews. For his TeX-nical expertise, I am grateful to Dr Tony Samuel. I extend thanks to Prof Mariusz Urbanski´ for introducing me to the world of dynamical systems in the first place. Last, but by no means least, I would like to thank Tushar Das for being a great friend and support, mathematically and otherwise. To my wonderful partner Andrew, my gratitude knows no bounds. My parents and my sister Jane (and her lovely boyfriend) have helped to keep me sane throughout the last four years; I dedicate the work in this thesis to them all. Extra thanks are due to my Dad, for proofreading my entire thesis. This was definitely above and beyond the call of duty. The moral support given to me by many friends has been invaluable. I would particularly like to mention Cecilia, Margaret, Karen and Crawford. Finally, it gives me great sorrow that my friend David is no longer here to share in my happiness at finishing my PhD. I also dedicate this thesis to his memory. vi Contents Introduction ix 1 Preliminaries 1 1.1 Hausdorff measure and dimension . 1 1.2 Dynamical systems . 3 1.2.1 Examples of number-theoretic dynamical systems . 5 1.3 The classical Lüroth expansion . 9 2 Introduction to a-Farey and a-Lüroth maps 15 2.1 Linearised generalisations of the Gauss map . 15 2.2 Linearised Farey-like maps . 21 2.3 Topological Properties . 28 2.4 Expanding and expansive partitions . 36 2.5 Hausdorff dimension of Diophantine-type sets . 41 2.5.1 Good-type sets for the a-Lüroth system . 43 2.5.2 Strict Jarník sets for the a-Lüroth system . 50 2.6 Appendix to Chapter 2: The non-alternating case . 55 3 Ergodic theoretic properties of Fa and La 57 3.1 Measure and ergodic theoretic preliminaries . 57 3.2 Ergodic theoretic properties of La ......................... 60 3.3 Ergodic properties of Fa .............................. 63 4 Renewal Theory 71 4.1 Classical renewal results . 71 4.2 Renewal results applied to a-sum-level sets . 76 5 Lyapunov spectra for Fa and La 83 5.1 Introduction to multifractal analysis . 83 5.2 General multifractal results of Jaerisch and Kesseböhmer . 86 5.3 Main Theorems . 90 5.4 Examples . 101 vii viii CONTENTS 6 Good sets and strict Jarník sets 109 6.1 Hyperbolic Geometry preliminaries . 109 6.1.1 The Poincaré disc and upper half-plane models . 109 6.1.2 The cross-ratio and some hyperbolic distance estimates . 113 6.1.3 Triangles and circles . 114 6.1.4 Classification of isometries . 116 6.1.5 Fuchsian Groups . 117 6.1.6 The Limit Set of a Fuchsian Group .
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