Some problems in ergodic theory
The thesis consists of a study of problems in ergodic theory relating to one-dimensional dynamical systems, Markov chains and generalizations of Markov chains. It is divided into chapters, three of which have appeared in the literature as papers. Chapter 1 looks at continuous families of circle maps and investigates conditions under which there is a weak*-continuous family of invariant measures. Sufficient conditions are exhibited and the necessity of these conditions is investigated. Chapter 2 is about expanding maps of the interval and the circle, and their relation with g-measures and generalized baker's transformations. The g-measures are generalizations of Markov chains to stochastic processes with infinite memory and generalized baker's transformations are geometric realizations of these. The chapter is based around the question of whether there exist expanding maps preserving Lebesgue measure, for which Lebesgue measure is not ergodic. Results are known if the map is sufficiently differentiable (for example C1+α), but the C1 case is still unclear. The chapter contains some partial solutions to this question. Chapter 3 is about representation of Markov chains on compact manifolds by measured collections of smooth maps. Given a measured collection of maps, a Markov chain is induced in a natural fashion. This chapter is about reversing this process. Chapter 4 describes a specialization of the setting of Chapter 3 to Markov chains on tori. In this case, it is possible to demand more of the maps of the representation than smoothness. In particular, they can be chosen to be local diffeomorphisms. The chapter also addresses the question of whether in general the maps can be taken to be diffeomorphisms and gives a counterexample showing that there exist Markov chains on tori which do not admit a representation by diffeomorphisms.