Some Studies Of Sensitivity In Topological Dynamical Systems

Given a dynamical system, to study its complexity is a centeral problem in thestudies of dynamical systems. Chaos as an important indicator to describe thecomplexity of a dynamical system was initially introduced by T. Li and J. Yorke in[26]. From then on, it’s known as Li-Yorke chaos, and the researches on chaos areendless, one of the most popular chaos is Devaney chaos which was introduced byDevaney in [27]. In the definition of Devaney, the sensitive dependence on initialconditions is understood as the nature part. In this thesis, we mainly study thesensitivity of a dynamical system. This thesis is divied into five chapters.In chapter one, we give a review about the historical background and someknown achievements which are important for this thesis in topological dynamicalsystems.In chapter two, we discuss N sensitivity in dynamical systems, and introducesome new concepts of N Syndetic sensitivity, N ergodic sensitivity, N Banachsensitivity, then prove that f has N Syndetic sensitivity, N ergodic sensitivityor N Banach sensitivity if the minimal points, quasi-weakly almost periodic pointsor positive Banach upper density points of f are dense and f is N sensitive.In chapter three, we discuss the sensitivity of uniformly convergence mappings.Connecting F sensitivity with uniformly convergence and measure center, viaFurstenberg families, we study the relationship between the F sensitivity of asequence of strongly uniformly convergence mappings and that of the limit mapping.By using the result that the set of quasi-weakly almost periodic points is dense inmeasure center and some known properties of quasi-weakly almost periodic points,we prove that the limit mapping of a sequence of strongly uniformly convergencemappings with F sensitivity is F sensitivity. If each mapping has a full measurecenter, then the limit mapping also has a full measure center.In chapter four, we discuss the sensitivity of flows, we prove that if issensitive and has a full measure centre, then is ergodic sensitive; if is sensitive and the set of minimal points of is dense, then is Syndetic sensitive.In chapter five, we consider the equivalent definitions of (quasi) weakly almostperiodic point on flows, we prove that if is sensitive and has a full measure centre,then is ergodic sensitive, while the closure of (quasi) weakly almost periodicpoints is the measure center, so the studies about equivalent conditions of (quasi)weakly almost periodic points on the flows in this chapter are valuable.