Existence and uniqueness of equilibrium states for systems with specification at a fixed scale: an improved Climenhaga–Thompson criterion*

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. This improvement will be crucial in a subsequent work, where we will prove that (open and densely) every Lorenz attractor supports a unique measure of maximal entropy.

Measures of maximal entropy of bounded density shifts

“…In the second part of the following proposition, we use techniques from Misiurewicz's proof of the variational principle [16] to build measures with entropy higher or equal than that of a sub-language. These applications of the tools from [16] have already been noted in [4, Proposition 5.1] and [17,Lemma 6.8]. PROPOSITION 3.3.…”

“…Remark. Using results from [17], Climenhaga explained in a blog post [7] that condition (3) is actually not required to prove uniqueness of the measure of maximal entropy.…”

“…which is obtained using the definition of weak* convergence. Then, combining equations ( 16) and (17) yields…”