Variational principles for pointwise preimage entropies

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