Relative topological conditional entropy and a Ledrappier's type variational principle for it

Abstract

In this paper we provide a sufficient condition for the existence of invariant measures with maximal relative measure-theoretic entropy, by introducing a new invariant for any factor map between topological dynamical systems, the concept of relative topological conditional entropy. It is proved a Ledrappier's type variational principle concerning relative topological conditional entropy. Consequently, if the factor map has zero relative topological conditional entropy (such a factor map is called asymptotically $ h $-expansive), then there exist invariant measures with maximal relative measure-theoretic entropy (i.e., whose relative measure-theoretic entropy equals exactly relative topological conditional entropy of the factor map). We explore further properties of relative topological conditional entropy, for example, the addition formula for a product factor map, the estimation of it for composition of two factor maps, and so on. We also interpret relative topological conditional entropy, respectively, using relative topological entropy of subsets and Bowen's dimensional entropy.