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arXiv · 2305.09835

Invariant measures of Toeplitz subshifts on non-amenable groups

Paulina Cecchi Bernales·María Isabel Cortez·Jaime Gómez
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Abstract

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we construct a Toeplitz $G$-subshift $(X,\sigma,G)$, which is an almost one-to-one extension of $\overleftarrow{G}$, having $r$ ergodic measures $\nu_1, \cdots,\nu_r$ such that for every $1\leq i\leq r$ the measure-theoretic dynamical system $(X,\sigma,G,\nu_i)$ is isomorphic to $\overleftarrow{G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups), however, we point out the differences and obstructions that could appear when the acting group is not amenable.

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Paulina Cecchi Bernales, María Isabel Cortez, Jaime Gómez. 2023-05-16. Invariant measures of Toeplitz subshifts on non-amenable groups. https://doi.org/10.1017/etds.2024.16

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