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A local variational principle for conditional entropy

Published online by Cambridge University Press:  13 January 2006

WEN HUANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail: wenh@mail.ustc.edu.cn, ghzhang@mail.ustc.edu.cn, yexd@ustc.edu.cn)
XIANGDONG YE
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail: wenh@mail.ustc.edu.cn, ghzhang@mail.ustc.edu.cn, yexd@ustc.edu.cn)
GUOHUA ZHANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail: wenh@mail.ustc.edu.cn, ghzhang@mail.ustc.edu.cn, yexd@ustc.edu.cn)

Abstract

For a given factor map $\pi:X\longrightarrow Y$ between two topological dynamical systems and a Borel cover ${\mathcal U}$, two notions of measure-theoretical conditional entropy $h_\mu^+(T,{\mathcal U}\mid Y)$ and $h_\mu^-(T,{\mathcal U}\mid Y)$ for an invariant Borel probability measure $\mu$ are introduced. It is shown that $h_\mu^+(T,{\mathcal U}\mid Y)=h_\mu^-(T,{\mathcal U}\mid Y)$. Moreover, $\max_{\mu}h_\mu^+(T,{\mathcal U}\mid Y)=h_{{\rm top}}(T,{\mathcal U}\mid Y)$ when $\mathcal U$ is an open cover. The relative variational principle is a consequence of the results.

Type
Research Article
Copyright
2006 Cambridge University Press

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