On non-uniform specification and uniqueness of the equilibrium state in expansive systems

TL;DR

This paper extends Bowen's classical results on the uniqueness of equilibrium states in expansive systems by introducing non-uniform conditions with sublogarithmic growth, proving uniqueness under these generalized hypotheses.

Contribution

It generalizes Bowen's theorem to non-uniform specification and transitivity with sublogarithmic growth functions, establishing new conditions for uniqueness of equilibrium states.

Findings

Uniqueness of equilibrium state holds under non-uniform hypotheses with sublogarithmic growth.

The unique equilibrium state has the K-property in certain cases.

Examples demonstrate the applicability of the generalized results.

Abstract

In [2], Bowen showed that for an expansive system (X, T) with specification and a potential with the Bowen property, the equilibrium state is unique and fully supported. We generalize that result by showing that the same conclusion holds for non-uniform versions of Bowen's hypotheses in which constant parameters are replaced by any increasing unbounded functions f(n) and g(n) with sublogarithmic growth (in n). We prove results for two weakenings of specification; the first is non-uniform specification, based on a definition of Marcus in ([14]), and the second is a significantly weaker property which we call non-uniform transitivity. We prove uniqueness of the equilibrium state in the former case under the assumption that liminf (f(n) + g(n))/ln n = 0, and in the latter case when lim (f(n) + g(n))/ln n = 0. In the former case, we also prove that the unique equilibrium state has the K-property. It is known that when f(n)/ln n or g(n)/ln n is bounded from below, equilibrium states may not be unique, and so this work shows that logarithmic growth is in fact the optimal transition point below which uniqueness is guaranteed. Finally, we present some examples for which our results yield the first known proof of uniqueness of equilibrium state.