## Taiwanese Journal of Mathematics

### ESTIMATE FOR SUPREMUM OF CONDITIONAL ENTROPY ON A CLOSED SUBSET

Wen-Chiao Cheng

#### 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

#### 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

#### References

• R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.
• Wen-Chiao Cheng, Variational Inequality Relative to a Closed Invariant Subgroup, Far East Journal of Dynamical Systems, 9(1) (2007), 1-16.
• Wen-Chiao Cheng and Sheldon Newhouse, Pre-image entropy, Ergodic Theory and Dynamical Systems, 25 (2005), 1091-1113.
• M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Spring Lecture Notes in Math. 527, Spring: New York, 1976.
• T. Downarowicz and J. Serafin, Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math., 172 (2002), 217-247.
• Anatole Katok and Boris Hasselblatt, An introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and Its Applications, Vol. 54, Cambridge University Press, Cambridge, 1995.
• Wen Huang, Xiangdong Ye and Guohua Zhang, A local variational principle for conditional entropy, Ergodic Theory and Dynamical Systems, 26 (2006), 219-245.
• K. Petersen, Ergodic Theory, Cambridge University Press, 1981.
• V. A. Rohlin, On the Fundamental Ideals of Measure Theory, American Mathematical Society, Translation Number 71, 1952.
• P. Walters, An Introduction to ergodic theory, Springer Lecture Notes, 458 (1982).