On topological and measure-theoretic properties of balanced symbolic dynamical systems
- 주제(키워드) Symbolic dynamics , Shift space , Subshift , Gibbs measure , Balanced property , Almost specification
- 주제(DDC) 510
- 발행기관 아주대학교 일반대학원
- 지도교수 최수영
- 발행년도 2024
- 학위수여년월 2024. 8
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/ajou/000000034074
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
Symbolic dynamics is the study of shift spaces. 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. 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. 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 $f$.
more목차
1 Introduction 1
1.1 Introduction 1
1.2 Preliminary 4
2 Relations between the balanced properties and the almost specification property 10
2.1 One-sided balanced property 10
2.2 Equivalence of the almost specification and the bi-balanced properties 22
3 Existence of invariant Gibbs measures 31
3.1 A relation between invariant Gibbs measures and the bi-balanced property 31
