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Pre-image pressure and invariant measures

Published online by Cambridge University Press:  12 February 2007

FANPING ZENG
Affiliation:
Institute of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China (e-mail: fpzeng@gxu.edu.cn) Department of Mathematics, Liuzhou Teachers College, Liuzhu, Guangxi 545004, People's Republic of China
KESONG YAN
Affiliation:
Institute of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China (e-mail: fpzeng@gxu.edu.cn) Department of Mathematics, Liuzhou Teachers College, Liuzhu, Guangxi 545004, People's Republic of China
GENGRONG ZHANG
Affiliation:
Institute of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China (e-mail: fpzeng@gxu.edu.cn)
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Abstract

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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.

Type
Research Article
Copyright
2007 Cambridge University Press