Entropy via preimage structure

Abstract

To measure the complexity of a “non-invertible” continuous map f on a compact metric space via the preimage structure, Hurley introduced the notion of pointwise topological preimage entropies hm(f) and hp(f) in 1995. A natural question was: Can one introduce the counterpart of hm(f) or hp(f) from the measure-theoretic point of view for an f-invariant measure μ and obtain properties analogous to that for the classical entropies? Cheng and Newhouse made the first step on this topic and introduced the notions of preimage entropies hpre(f) and hpre,μ(f) and investigated their properties in 2005. Recently, Wu and Zhu proposed a notion of pointwise metric preimage entropy hm,μ(f) and obtained some properties for f with uniform separation of preimages.In this paper, we give a complete answer to the above question for general continuous maps. The Shannon-McMillan-Breiman theorem for hm,μ(f) and the variational principle relating hm,μ(f) and hm(f) are obtained. It is shown that hm(f)=hpre(f) and the three types of measure-theoretic entropy-like invariants hpre,μ(f), hm,μ(f) and the folding entropy Fμ(f) (introduced by Ruelle) via the preimage structure coincide with each other. We also show that the preimage entropy variational principle can be realized as a variant of the conditional variational principle. These results answer several open questions asked by Cheng-Newhouse and Downarowicz. The upper semi-continuity of hm,μ(f) with respect to invariant measures with the same separation rate is obtained. Moreover, the preimage pressure and preimage equilibrium states are investigated.