Ergodic measures of intermediate entropies for amenable group actions with an approximate product property

Wenda Zhang; Xiankun Ren

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arXiv · 2610.09949

Ergodic measures of intermediate entropies for amenable group actions with an approximate product property

Wenda Zhang·Xiankun Ren

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Abstract

We study entropy realization for continuous actions of infinite countable amenable groups. We introduce an approximate product property for amenable group actions that permits a small proportion of tracing mistakes on prescribed pairwise disjoint, sufficiently invariant finite sets. We prove that this property implies entropy-denseness and almost entropy-approximability of every invariant measure. The construction combines zero-entropy exact tilings with a finite-block estimate for the complexity of tracing mistakes. Under asymptotic entropy expansiveness, almost entropy-approximability upgrades to entropy-approximability. For every $0\leqα<h(X,G)$, ergodic measures of entropy $α$ then form a residual subset of the invariant measures whose entropy is at least $α$. In particular, the set of ergodic measure entropies equals $[0,h(X,G)]$.37A35, 37B40, 37B05

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Wenda Zhang, Xiankun Ren. 2026-10-07. Ergodic measures of intermediate entropies for amenable group actions with an approximate product property. https://arxiv.org/abs/2610.09949

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