Year: 2023
Type: preprint
Abstract: Let $G$ be a non-amenable countable group. We show that every almost automorphic $G$-action on a compact Hausdorff space, with a maximal equicontinuous factor whose phase space is a Cantor set, admits invariant probability measures (this partially answers a question posed by Veech). In particular, every Toeplitz $G$-subshift has a non-empty space of invariant measures, meaning that this family of subshifts is not a test for amenability for countable groups. We prove that almost one-to-one extensions without measures ensure the existence of symbolic almost one-to-one extensions with equal characteristics. As a consequence, we obtain the most general result of this paper. Finally, as a corollary of our results, we deduce that the class of Toeplitz subshifts is not dense in the space of infinite transitive subshifts of $Σ^G$, unlike $G=\mathbb{Z}$.
Source: arXiv (Cornell University)
Authors: Cortez, María Isabel, Gómez, Jaime
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4390091696