On controlled specification and uniqueness of the equilibrium state in expansive systems
In Bowen (1974/75 Math. Syst. Theory 8 193-202), Bowen showed that for an expansive system 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 weakened versions of Bowen’s hypotheses in which constant parameters are replaced by any increasing unbounded functions and with sublogarithmic growth (in n). We prove results for two weakenings of specification; the first is controlled specification, based on a definition of Marcus (1980 Mon.hefte Math. 89 121-9), and the second is a significantly weaker property which we call controlled transitivity. We prove uniqueness of the equilibrium state in the former case under the assumption that , and in the latter case when . In the former case, we also prove that the unique equilibrium state has the K-property. It is known that when or 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.