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arXiv · 2605.21048

Ergodic measures of intermediate entropies for $\mathbb{Z}^{d}$-action

Yage Liu·Ercai Chen·Xiaoyao Zhou
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Abstract

For dynamical systems satisfying the approximate $\mathbb{Z}^{d}$ or $\mathbb{Z}_+^{d}$-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In particular, we prove that the set of ergodic measures with any given intermediate entropy is generic in certain natural subspaces. As a consequence, this result confirms Katok's conjecture on the existence of ergodic measures with arbitrary intermediate entropy for such systems.

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Yage Liu, Ercai Chen, Xiaoyao Zhou. 2026-05-20. Ergodic measures of intermediate entropies for $\mathbb{Z}^{d}$-action. https://arxiv.org/abs/2605.21048

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