Ergodic measures of intermediate entropies for amenable group actions with an approximate product property
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≤α<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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BibTeX
@misc{zhang2026ergodic,
title = {{Ergodic measures of intermediate entropies for amenable group actions with an approximate product property}},
author = {Wenda Zhang and Xiankun Ren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09949},
eprint = {2610.09949},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09949},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ergodic measures of intermediate entropies for amenable group actions with an approximate product property
AU - Wenda Zhang
AU - Xiankun Ren
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09949
UR - https://arxiv.org/abs/2610.09949
AB - 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≤α<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
ER -