Positive entropy equilibrium states

TL;DR

This paper establishes conditions under which equilibrium states in certain dynamical systems have positive entropy, extending known results to non-uniform specification scenarios.

Contribution

It introduces a non-uniform specification condition ensuring positive entropy of equilibrium states, and shows this does not hold for other non-uniform specifications.

Findings

Positive entropy equilibrium states under the new condition.

Counterexamples for other non-uniform specifications.

Extension of classical results to broader settings.

Abstract

For transitive shifts of finite type, and more generally for shifts with specification, it is well-known that every equilibrium state for a Holder continuous potential has positive entropy as long as the shift has positive topological entropy. We give a non-uniform specification condition under which this property continues to hold, and demonstrate that it does not necessarily hold for other non-uniform versions of specification that have been introduced elsewhere.