On topological and measure-theoretic properties of balanced symbolic dynamical systems

Abstract

Symbolic dynamics is the study of shift spaces._x000D_ <br>The balanced properties of a shift space are combinatorial properties of words. The purpose of this thesis is to study of topological and measure-theoretic properties of balanced shift spaces._x000D_ <br>In Chapter 2, relations between the balanced properties and the almost specification property are given. We construct two types of one-sided balanced shift spaces and show that the one-sided balanced property and the almost specification property are not equivalent. In the class of coded systems, a condition for the word entropy of the collection of subwords of generators of given coded system implies the equivalence of the bi-balanced property and the almost specification property._x000D_ <br>In Chapter 3, we extend the notion of the balanced properties using weighted sums scaled by a real-valued continuous function on a shift space and find a connection between the existence of invariant Gibbs measures for a real-valued continuous function $f$ on a shift space $X$ and the bi-balanced property of $X$ with respect to $f$. It is proven that a shift space $X$ is bi-balanced with respect to a real-valued continuous function $f$ on $X$ if and only if it has an invariant Gibbs measure for $f$.