Equilibrium states of intermediate entropies

Abstract

We explore a new approach to the conjecture of Katok on intermediate entropies, which shows that the conjecture can be a consequence of certain results on equilibrium states. Suppose that the system has finite topological entropy and the entropy function is upper semi-continuous. Let be a continuous potential such that has a unique equilibrium state for every and the maximizing measures of have zero entropy. Then the entropies of these equilibrium states can assume any intermediate value between zero and the topological entropy of the system. As an application, we prove Katok's conjecture for Mañé diffeomorphisms.