Pre-image pressure and invariant measures

Abstract

We define and study a new invariant called pre-image pressure and its relationship with invariant measures. More precisely, for a given dynamical system $(X,f)$ (where $X$ is a compact metric space and $f$ is a continuous map from $X$ to itself) and $\varphi\in C(X,R)$ (the space of real-valued continuous functions on $X$), we prove a variational principle for pre-image pressure $P_{\rm pre}(f,\varphi),\ P_{\rm pre}(f,\varphi) = \sup_{\mu \in \mathcal{M}(f)} \{ h_{{\rm pre},\mu}(f) + \int \varphi \,d\mu\}$, where $h_{{\rm pre},\mu}(f)$ is the pre-image entropy (W.-C. Cheng and S. Newhouse. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113) and $\mathcal{M}(f)$ is the set of invariant measures of $f$. Moreover, we also prove that pre-image pressure determines the invariant measures and give some applications of pre-image pressure to equilibrium states.