Measures of maximal relative entropy with full support

Abstract

Let pi be a factor map from an irreducible shift of finite type X to a shift space Y. Let nu be an invariant probability measure on Y with full support. We show that every measure on X of maximal relative entropy over nu is fully supported. As a result, given any invariant probability measure nu on Y with full support, there is an invariant probability measure mu on X with full support that maps to nu under pi. If nu is ergodic, mu can be chosen to be ergodic. These results can be generalized to the case of sofic shifts. We demonstrate that the results do not extend to general shift spaces by providing counterexamples.