One-parameter families of measure-theoretic and topological entropies

Abstract

We consider one-parameter families of measure-theoretic and topological entropies associated with dynamical systems. These families are such that, for a particular parameter value, well stablished quantities in Ergodic Theory are recovered. Concerning the family of measure-theoretic entropies, which was introduced by the authors in a previous work, we review the definition and some of its properties, particularly the isomorphism theorems, and also present new results and perform calculations for some particular dynamical systems. With respect to the topological entropies two families, which are shown to be equivalent for a set of values of their parameter, are defined and their properties, such as the topological invariance, studied. We also outline the relationship between the members of both, the topological and the measuretheoretic, families. Finally, within the same spirit, we introduce families of topological pressures.