Taiwanese Journal of Mathematics


Dongkui Ma and Bin Cai

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By using the Carathéodory-Pesin structure (C-P structure), the topological entropy on the whole space introduced for a proper map, is generalized to the cases of arbitrary subset, i.e., we introduce three notions of topological entropy. Some of the properties of these notions are provided. As some applications, for the proper map of locally compact separable metric space, we prove the following variational principles: (1) The upper capacity topological entropy on any subset and the minimum of the Bowen-Dinaburg entropies always coincide; (2) For any invariant probability measure, the measure-theoretic entropy and the infimum of the topological entropies on all sets which are of full measures always coincide; (3) The relationship between the topological entropies of level sets of the ergodic average of some continuous functions and the measure-theoretic entropies are given. These are the extensions of results of Patrão and Pesin, etc.

Article information

Taiwanese J. Math., Volume 18, Number 4 (2014), 1219-1241.

First available in Project Euclid: 10 July 2017

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Zentralblatt MATH identifier

Primary: 37A35: Entropy and other invariants, isomorphism, classification 37B40: Topological entropy 37C45: Dimension theory of dynamical systems

proper map admissible cover C-P structure topological entropy variational principle


Ma, Dongkui; Cai, Bin. TOPOLOGICAL ENTROPY OF PROPER MAP. Taiwanese J. Math. 18 (2014), no. 4, 1219--1241. doi:10.11650/tjm.18.2014.3339. https://projecteuclid.org/euclid.twjm/1499706486

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