Eli Glasner

Ergodic Theory via Joinings

Introduction General group actions: Topological dynamics Dynamical systems on Lebesgue spaces Ergodicity and mixing properties Invariant measures on topological systems Spectral theory Joinings Some applications of joinings Quasifactors Isometric and weakly mixing extensions The Furstenberg-Zimmer structure theorem Hosts theorem Simple systems and their self-joinings Kazhdans property and the geometry of

Sensitive dependence on initial conditions

M_{\Gamma}(\mathbf{X})

WEAK ORBIT EQUIVALENCE OF CANTOR MINIMAL SYSTEMS

Entropy theory for

On minimal actions of Polish groups

\mathbb{Z}

Residual properties and almost equicontinuity

-systems: Entropy Symbolic representations Constructions The relation between measure and topological entropy The Pinsker algebra, CPE and zero entropy systems Entropy pairs Kriegers and Ornsteins theorems Prerequisite background and theorems Bibliography Index of symbols Index of terms.

Chapter 10 - On the Interplay between Measurable and Topological Dynamics

It is shown that the property of sensitive dependence on initial conditions in the sense of Guckenheimer follows from the other two more technical parts of one of the most common recent definitions of chaotic systems. It follows that this definition applies to a broad range of dynamical systems, many of which should not be considered chaotic. We investigate the implications of sensitive dependence on initial conditions and its relation to dynamical properties such as rigidity, ergodicity, minimality and positive topological entropy. In light of these investigations and several examples which we exhibit, we propose a natural family of dynamical systems- chi -systems-as a better abstract framework for a general theory of chaotic dynamics.

Minimal actions of the group \( {\Bbb S(Z)} \) of permutations of the integers

This paper is a commentary on the recent work [4]. It has two goals: the first is to eliminate the C*-algebra machinery from the proofs of the results of [4]; the second, to provide a characterization of weak orbit equivalence of Cantor minimal systems in terms of their dimension groups.

Quasi-factors of zero-entropy systems

Abstract We show the existence of an infinite monothetic Polish topological group G with the fixed point on compacta property. Such a group provides a positive answer to a question of Mitchell who asked whether such groups exist, and a negative answer to a problem of R. Ellis on the isomorphism of L(G), the universal point transitive G-system (for discrete G this is the same as βG the Stone-Cech compactification of G) and E(M,G), the enveloping semigroup of the universal minimal G-system (M,G). For G with the fixed point on compacta property M is trivial while L(G) is not. Our next result is that even for Z with the discrete topology, L(Z) = βZ is not isomorphic to E(M, Z). Finally we show that the existence of a minimally almost periodic monothetic Polish topological group which does not have the fixed point property will provide a negative answer to an old problem in combinatorial number theory.

A variation on the variational principle and applications to entropy pairs

A propertyP of a compact dynamical system (X,f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all propertyP systems is a residual property providedP is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from allP systems iff the onlyP systems upon which Λ acts are trivial. We use this to prove that every monothetic group has an action which is weak mixing and topologically ergodic.

Local entropy theory

This chapter discusses the interplay between measurable and topological dynamics. Ergodic theory or measurable dynamics and topological dynamic are the two sister branches of the theory of dynamical systems. The simplest dynamical systems are the periodic ones. In the absence of periodicity, the crudest approximation to this is approximate periodicity. In the topological setup, there is no such convenient decomposition describing the system in terms of its indecomposable parts and one has to use some less satisfactory substitutes. Natural candidates for indecomposable components of a topological dynamical system are the “orbit closures” (i.e. the topologically transitive subsystems) or the “prolongation” cells that often coincide with the orbit closures. The minimal subsystems are of particular importance in this chapter. Although the study of the general system can be reduced to that of its minimal components, the analysis of the minimal systems is nevertheless an important step toward a better understanding of the general system.