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 $\nu _1, \ldots ,\nu _r$ 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.

中文翻译：

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Toeplitz subshifts 对不服从群体的不变测度

让G是可数残差有限群（例如， ${\mathbb F}_2$ ） 然后让 $\overleftarrow {G}$ 是完全断开的度量压缩G配备了行动G通过左乘法。对于每一个 $r\geq 1$ ，我们构造一个 ToeplitzG-subshift $(X,\西格玛,G)$ ，这几乎是一对一的扩展 $\overleftarrow {G}$ ，有r遍历测度 $\nu _1、\ldots、\nu _r$ 这样对于每个 $1\leq i\leq r$ ，测度论动力系统 $(X,\西格玛,G,\nu _i)$ 同构于 $\overleftarrow {G}$ 赋予哈尔测度。我们提出的构造是通用的（对于服从和不服从的剩余有限群）；然而，我们指出，当行动小组不听话时，可能会出现分歧和障碍。