Relativization And Localization Of Dynamical Properties

The thesis consists of two topics of relativization and localization of dynamical properties. In the first part, we generalize systematically to the relative case many developments obtained recently in topological dynamics, including the local entropy theory built in the past fifteen years. In the second part, on one hand, we introduce new methods of localization to more deeply understand the complexity of a dynamical system; on the other hand, we build the local entropy theory of a general group actions, especially a countable discrete amenable group actions, on a compact metric space. Especially, we emphasize the parallelism between ergodic theory and topological dynamics and the applications of ergodic theory in the study of topological dynamics.The thesis is organized as follows:In Chapter 1, the origin, developments and main objective and contents of ergodic theory and topological dynamics are recalled, and then the backgrounds and main results of the thesis are summarized as three parts, respectively.In Chapter 2, some preliminary notions and results in ergodic theory and topological dynamics, which will be used in the thesis, are reviewed.Chapters 3, 4 and 5 focus on the first topic of the thesis, namely, the relativization of dynamical properties. Precisely speaking, letπ: (X, T)→(Y, S) be a factor map between dynamical systems, i.e. a continuous surjective mapping from X onto Y compatible with the actions on X and Y, eachπ^-1(y), y∈Y is called a fiber. Given a factor map between dynamical systems, we aim to study the dynamical behaviors on the fibers. In particular, if (Y, S) is a trivial dynamical system then the study of the fibers is just the absolute case of the study of (X, T).In Chapter 3, the correspondence of the complexity function and equicontinuity in the relative case, the relative complexity function and the positively equicontinuous extension, are studied. It is shown that, for a given open factor map between minimal dynamical systems, the factor map is positively equicontinuous if and only if each open cover has a bounded relative complexity function. Then based on the idea of the relative complexity function, the notions of relative complexity n-tuples, relative n-scattering (n≥2) and relative scattering are introduced. When considering an open factor map between minimal invertible dynamical systems, it is proved the existence of the maximal invertible equicontinuous factor of the given factor map and that relative scattering implies weak mixing.In Chapter 4, chaotic behaviors on the fibers, relative sensitivity and relative My-cielski chaos, are introduced. First, the notion of relative sensitivity is introduced. It is proved that: for a factor map between minimal dynamical systems, it is either relative sensitive or positively equicontinuous; any non-trivial weakly mixing factor map between dynamical systems is relative sensitive. The known Glasner-Weiss result tells us that any M-system, which is not minimal, is sensitive. It is generalized to the relative case. Meanwhile, it is proved that, in many cases, relative 2-scattering implies relative Myciel-ski chaos. Then, when considering a factor map between invertible dynamical systems, it is shown that the positivity of relative entropy implies the existence of proper asymptotic pairs on fibers and relative Mycielski chaos. Moreover, it turns out that the scrambled subsets on fibers are topologically very big, i.e. the supremum of the topological entropy of the scrambled subsets on fibers equals the relative entropy of the given factor map. Even in the absolute case, this is a new and deep result.In Chapter 5, the local entropy theory of a dynamical system is generalized completely to the relative case. Precisely speaking, the relative local variational principle of a finite open cover is established. Then as its application, the notion of relative entropy tuples is introduced both in topological and measure-theoretic settings. The variational relation between these two kinds of relative entropy tuples is interpreted. And it is proved the existence of relative topological Pinsker factor, which answers affirmatively some question mentioned in [111]. Moreover, based on the idea of relative topological entropy pairs, the notion of relative uniformly positive entropy and relative completely positive entropy extensions is introduced. Some basic properties and the finite production is studied. In the process, on one hand, it seems that the key lemma [12, Lemma 1] is very difficult to be generalized to the relative case, and so some new method is needed; on the other hand, in the measure-theoretic setting two kinds of relative entropy for a finite Borel cover are introduced and turn out to be the same. Note that the equivalence of these two kinds entropy will play an important role in the building of the local entropy theory for a countable discrete amenable group actions on a compact metric space, and the equivalence of them for a finite open cover has just been obtained recently.Chapters 6 and 7 aim to another topic of the thesis, namely, the localization of dynamical properties.In Chapter 6, in order to more deeply understand the complexity of a dynamical system, some new methods of localization are applied. On one hand, along the line of entropy pairs, tuples, sequences and sets, the notion of C-entropy point both in topological and measure-theoretic settings is introduced. The structure of the set of C-entropy points is studied, and the variational relation between these two kinds of C-entropy points is established. On the other hand, using the idea of Bowen's separated and spanning subsets, the notion of uniformly entropy point is introduced and studied. A maybe unexpected byproduct of the study of uniformly entropy point is that, in each dynamical system, there exists a countable closed subset such that its topological entropy equals the topological entropy of the origin system. Note that it is known that each homeo-morphism on a compact countable space has zero topological entropy. Moreover, more interesting and deep results are obtained in [82].In Chapter 7, the local entropy theory of a countable discrete amenable group actions, which is more general than (?) actions, on a compact metric space is built, including all countable discrete solvable group actions. This generalizes completely to this general group actions the local entropy theory set up in the past fifteen years (see for example a survey being prepared by Profs. E. Glasner and X. Ye [57]). First, the local variational principle of a finite open cover is established in this setting, which can be used to obtain the classic variational principle by some standard arguments. Then, as its application, the notion of entropy tuples both in topological and measure-theoretic settings is introduced, the structure of the set of entropy tuples is studied and the variational relation between these two kinds of entropy tuples is interpreted. Finally, based on the idea of topological entropy pairs, topological analogue of Kolmogorov systems in ergodic theory, namely uniformly positive entropy and completely positive entropy, are introduced and studied. It turns out that: uniformly positive entropy implies some strong transitivity properties, completely positive entropy implies the existence of invariant measures with full support and they are both preserved under the finite production. In the establishing of the local variational principle, in the measure-theoretic setting two kinds of entropy for a finite Borel cover are introduce, studied and turn out to be the same. Whereas, different from the case of (?) actions, to prove the upper semi-continuity of these two kinds of entropy with respect to invariant measures, it is discovered some deep property underlying a countable discrete amenable group; to prove the equivalence of them, besides of the results obtained in Chapter 5, it becomes inevitable the orbital pointview of entropy theory in this setting developed by A. I. Danilenko in [23].