On Measurable Sensitivity And Related Problems Of Abelian Group Actions

Chaos is a very important field in the study of dynamical system, among various definitions of chaos, the Devaney’s chaos which has sensitivity as its core, is very important. Recently, a lot of progress on sensitivity have been made. However most of results are investigated under Z (or Z+) actions, and the research of general group actions is not much. In this thesis, we let the acting group of a topological dynamical system be a countable discrete abelian group and we introduce the notion of measurable sensitivity and study some of its properties. The thesis is organized as follows:In Chapter 1, some related backgrounds and recently progresses are introduced, and we give a brief introduction about what we have done in this thesis.In Chapter 2, we introduce some of related notions, theories and conclusions.In Chapter 3, the notion of measurable sensitivity is introduced and we give a chara-cterization of a minimal system which is measurable n-sensitive but not measurable (n+1)-sensitive, i.e. let π be a factor map from minimal system (X, G) to its maximal equicontinuous factor (Y,G) and μ∈M(X,G), v=πμ, then system (X,G) is n-sensitive but not (n+1)-sensitive for μ if and only if max{| π-1y|:y∈ Y}= n (n≥2).In Chapter 4, we define some notions under countable discrete abelian group actions, such as measurable pairwise sensitivity, measurable expansivity and measurable equico-ntinuity. We obtain a sufficient condition of measurable pairwise sensitivity which is measurable weakly mixing and proves that measurable expansivity is equivalent to measurable pairwise sensitivity and measurable sensitivity is equivalent to measurable pairwise sensitivity for a invariant Borel probability measure. Moreover, we discuss the relationship between measurable pairwise sensitivity and measurable equicontinuity, i.e. let (X,G) be a topological dynamical system and μ∈M(X,G), then{X,G) is not μ-equicontinuous if(X,G) is μ-pairwise sensitive.In the last Chapter, we give a summary for this paper and list several questions which to be solved.