Quasi-uniform Convergence in Dynamical Systems Generated by an Amenable Group Action

TL;DR

This paper investigates the properties of the Weyl pseudometric in dynamical systems generated by amenable group actions, establishing continuity and semicontinuity results for entropy, minimal subsets, and invariant measures, and applying these to Toeplitz sequences.

Contribution

It introduces the notion of regular Toeplitz sequences for amenable residually finite groups and proves their minimality and unique ergodicity, providing a new proof regarding entropy values.

Findings

Topological entropy is lower semicontinuous with respect to the Weyl pseudometric.

The simplex of invariant measures varies continuously under the Weyl pseudometric.

Regular Toeplitz sequences are minimal and uniquely ergodic, with path-connected families.

Abstract

We study properties of the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both lower semicontinuous with respect to the Weyl pseudometric. Furthermore, the simplex of invariant measures supported on the orbit closure varies continuously. We apply the Weyl pseudometric to Toeplitz sequences for amenable residually finite groups. We introduce the notion of a regular Toeplitz sequence for an arbitrary amenable residually finite group and demonstrate that all regular Toeplitz sequences define minimal and uniquely ergodic systems. We prove path-connectivity of this family with respect to the Weyl pseudometric. This leads to a new proof of a theorem of Fabrice Krieger, which says that the values of entropy of Toeplitz sequences can have arbitrary finite entropy.