Quasi-uniform convergence in dynamical systems generated by an amenable group action

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.