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

Abstract

We study 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 configurations for arbitrary amenable residually finite groups. We introduce the notion of a regular Toeplitz configuration and demonstrate that all regular Toeplitz configurations define minimal and uniquely ergodic systems. We prove that this family is path-connected in the Weyl pseudometric. This leads to a new proof of a theorem of Krieger, saying that Toeplitz configurations can have arbitrary finite entropy.is ergodic if for every G-invariant set A ⊂ X one has either μ(A) = 0 or μ(A) = 1. We denote by † We omit this assumption at the beginning of Section 7. ‡ We used the adjective to distinguish left Følner sequences from their right-and two-sided analogues. In a similar vain notions related to non-commutative groups have 'left' and 'right' variants. For brevity, we will use adjectives 'left/right' only in definitions and routinely omit them later, since the choice of 'left/right' is fixed throughout the paper.

Invariant measures of Toeplitz subshifts on non-amenable groups

“…In [25], the notion of regularity over Toeplitz subshifts is defined for the case where the acting group is amenable. Using the lemma below, it can be shown that the definition presented here coincides with that used in [25] when G is amenable. Note that, by definition, regular Toeplitz G-subshifts are uniquely ergodic.…”