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arXiv · 2209.01902

Non-existence of a universal zero entropy system via generic actions of almost complete growth

Georgii Veprev
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Abstract

We prove that a generic p.m.p. action of a countable amenable group $G$ has scaling entropy that can not be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of $G$ for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. $G$-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that can not be bounded from below by a given sequence. We also show an example of an amenable group that has such lower bound for every free p.m.p. action.

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Georgii Veprev. 2022-09-05. Non-existence of a universal zero entropy system via generic actions of almost complete growth. https://arxiv.org/abs/2209.01902

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