Taiwanese Journal of Mathematics

ESTIMATE FOR SUPREMUM OF CONDITIONAL ENTROPY ON A CLOSED SUBSET

Wen-Chiao Cheng

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Abstract

This paper compares the conditional metric entropy $h_{\mu}(T\mid G)$, with the topological entropy, $h_{top}(T\mid G)$, of a continuous map $T$, where $G$ is a closed fully $T$-invariant subset. The following Variational Inequality is proven, $$h_{top}(T\mid G)\leq \sup _{\mu\in M(X,T)}h_{\mu}(T\mid \langle G \rangle )\leq h_{top}(T\mid G)+h_{top}(T\mid cl(X\backslash G))$$ where $M(X,T)$ is the collection of all invariant measures of $X$, which is an extension of the usual variational principle when $G=X$.

Article information

Source
Taiwanese J. Math., Volume 12, Number 7 (2008), 1791-1803.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405089

Digital Object Identifier
doi:10.11650/twjm/1500405089

Mathematical Reviews number (MathSciNet)
MR2449666

Zentralblatt MATH identifier
1221.37027

Subjects
Primary: 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37B40: Topological entropy

Keywords
conditional metric entropy topological entropy variational inequality

Citation

Cheng, Wen-Chiao. ESTIMATE FOR SUPREMUM OF CONDITIONAL ENTROPY ON A CLOSED SUBSET. Taiwanese J. Math. 12 (2008), no. 7, 1791--1803. doi:10.11650/twjm/1500405089. https://projecteuclid.org/euclid.twjm/1500405089


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