Spectral and distributional problems for homogeneous extensions of dynamical systems and the rate of mixing of two-dimensional Markov shifts

Abstract

This thesis consists of four chapters. Chapters 1 and 2 are somewhat related in the sense that they deal with similar dynamical systems. Each chapter comes complete with its own references and notations. For the convenience of the reader, we provide an introduction and indeed an elongated summary to the whole thesis in Chapter 0. In Chapter 1, we study how closed orbits of a subshift of finite type hits to a finite homogeneous extension. In particular, we obtain an asymptotic formula for the number of closed orbits according to how they lift to the extension space. We apply our findings to the case of finite extensions and also to automorphism extensions of hyperbolic toral automorphisms. Chapter 2 deals with lifting ergodic properties of an arbitrary measure preserving transformation T to homogeneous extensions of T. Our results extends well known theorems already obtained for the case of compact group extensions of measure-preserving transformations. We also give simplified results to the special case when the base transformation is a Markov shift and the skewing-function depends on a finite number of coordinates. In Chapter 3, we look at the rate of mixing of rectangle sets of two dimensional Markov shifts with respect to the natural shift actions. We show that if one of the matrix defining the Markov measure is aperiodic then this rate is exponentially fast. We provide an example to illustrate what could happen in general