On Sensitivity, Sequence Entropy And Related Problems In Dynamical Systems

In this thesis, we mainly study the properties of sensitivity, sequence entropy andrelated problems in dynamical systems.The thesis is divided into 6 chapters and is organized as follows:In the Introduction, the origin, developments and main contents of the topological dynamical system and ergodic theory are presented.In Chapter 2, the basic notions and properties on topological dynamical system and ergodic theory are recalled.In Chapter 3, the properties of n-sensitivity, which was introduced by Xiong, are investigated, especially in the minimal case. It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct. Moreover, the structure of a minimal system which is n-sensitive but not (n + 1)-sensitive (n≥2) is determined.In Chapter 4, notions of sensitive sets (S-sets) and regionally proximal sets (Q-sets) are introduced. It is shown that a transitive system is sensitive if and only if there is an S-set with Card(S)≥2, and for a transitive system each S-set is a Q-set. Moreover, the converse holds when (X, T) is minimal. It turns out that each transitive (X, T) has a maximal almost equicontinuous factor. According to the cardinalities of the S-sets, transitive systems are divided into several classes. Characterizations and examples are given for this classification both in minimal and transitive non-minimal settings. It is proved that for a transitive system any entropy set is an S-set, and consequently, a transitive system which has no uncountable S-sets has zero topological entropy. Moreover, it is shown that a transitive, non-minimal system with dense set ofminimal points has an infinite S-set, and there exists a Devaney chaotic system which has no uncountable S-set. Finally, a non-minimal sensitive E-system is constructed such that each its S-set has cardinality at most 4.In Chapter 5, the notion of measurable n-sensitivity for measure preserving systems is introduced, and the relation between measurable n-sensitivity and the maximal pattern entropy is studied. It is proved that, when (X, B,μ, T) is ergodic and T is invertible, (X, B,μ, T) is measurable n-sensitive but not measurable n+1-sensitive if and only if h_μ^* (T) = log n, where h_μ^* (T) is the maximal pattern entropy of T.In the last Chapter, the properties of topological sequence entropy for TDSs on countable compact metric spaces are discussed. It is proved that when d(X)≤1, h_top^s(T) = 0; and when d(X)≥2, there exists a homeomorphism T on X such that X^d is the sequence entropy set of (X, T), where d(X) and X^d are the derived degree of X and the set of all accumulation points of X respectively.