Variational principles for pointwise preimage entropies

Abstract

Based on the preimage structure of the system , Hurley introduced the notion of pointwise topological preimage entropies and . Furthermore, from the measure-theoretic point of view, Wu and Zhu introduced a notion of pointwise metric preimage entropy for a T-invariant measure µ on X, and obtained the variational principle between and under the condition of uniform separation of preimages. A natural question is whether a variational principle for and without any additional assumptions. In this paper, we define a new version of topological preimage entropy relative to a T-invariant measure µ, and show that the inequality holds for every T-invariant probability measure µ. As a consequence, we obtain that there is a topological dynamical system such that the following strict inequality holds: where denote the set of all T-invariant probability measures.