Existence and uniqueness of equilibrium states for systems with specification at a fixed scale

TL;DR

This paper proves the uniqueness of equilibrium states for dynamical systems with weak, non-uniform specification properties at a fixed scale, extending classical results to broader conditions.

Contribution

It demonstrates that equilibrium states are unique under weak, non-uniform specification conditions on a limited set of orbit segments, generalizing previous uniform assumptions.

Findings

Equilibrium states are unique under weak specification at a fixed scale.

The approach extends classical Bowen and Franco methods to non-uniform settings.

Uniqueness holds even with specification only on a small collection of orbit segments.

Abstract

We consider the uniqueness of equilibrium states for dynamical systems that satisfy certain weak, non-uniform versions of specification, expansivity, and the Bowen property at a fixed scale. Following Climenhaga-Thompson's approach which was originally due to Bowen and Franco, we prove that equilibrium states are unique even when the weak specification assumption only holds on a small collection of orbit segments.