Mateusz Więcek (Wroclaw University of Science and Technology) скачать в хорошем качестве

Mateusz Więcek (Wroclaw University of Science and Technology)

Title: Asymptotic pairs in topological actions of countable amenable groups Abstract: By the known theorem of F. Blanchard, B. Host and S. Ruette, every topological Z-action of positive entropy admits asymptotic pairs. Moreover, T. Downarowicz and Y. Lacroix proved that every topological Z-action of entropy zero has an extension with no asymptotic pairs. Together, these two results give a characterization of zero-entropy topological Z-actions as factors of systems with no asymptotic pairs. Recently developed theory of multiorders allowed us to achieve a similar characterization for topological actions of countable amenable groups. We begin the talk by introducing the notion of a multiorder and present some basic properties of multiorders on countable amenable groups. Then, we provide a definition of a ≺-asymptotic pair in a topological action (X, G) of a countable amenable group G. In the case where for some G-invariant Borel probability measure μ on X, the measure-preserving system (X, μ, G) factors, via a map φ, onto some multiorder (O, ν, G), we also introduce the notion of a φ-asymptotic pair. Then we prove that if μ has positive measure-theoretic conditional entropy with respect to the multiorder factor, then the set of points which belong to φ-asymptotic pairs has positive measure μ. As a strengthening of this result, we show that for any system (X, G) of positive topological entropy, any multiorder (O, ν, G) and ν-almost every ≺ ∈ O, there exist ≺-asymptotic pairs in X. Both of these results generalizes the classical Blanchrd-Host-Ruette Theorem. Finally, we characterize systems (X, G) of topological entropy zero as factors of topologically multiordered systems with no φ-asymptotic pairs. The talk is based on the joint work with Tomasz Downarowicz.