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

TL;DR

This paper characterizes the structure of invariant measures in certain dynamical systems, confirming that ergodic measures with any intermediate entropy are generically present, thus validating a conjecture by Katok.

Contribution

It provides a detailed description of invariant measure spaces for systems with approximate product properties and confirms Katok's conjecture on intermediate entropy measures.

Findings

Ergodic measures with any intermediate entropy are generic in specific subspaces.

The results apply to systems satisfying approximate product properties and asymptotic entropy expansiveness.

Confirmed Katok's conjecture on the existence of intermediate entropy ergodic measures.

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.