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Slow Entropy for Noncompact Sets and Variational Principle

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Abstract

This paper defines and discusses the dimension notion of topological slow entropy of any subset for \({{\mathbb {Z}}}^d-\)actions. Also, the notion of measure-theoretic slow entropy for \({\mathbb {Z}}^d-\)actions is presented, which is modified from Brin and Katok (Geometric Dynamics, Springer, Berlin 1983). Relations between Bowen topological entropy Bowen (Trans Am Math, 184:125–136, 1973), and topological slow entropy are studied in this paper, and several examples of the topological slow entropy in a symbolic system are given. Specifically, a variational principle is proved.

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Acknowledgments

The authors would like to thank the anonymous referees for carefully reading our paper and providing suggestions for improvement. The work was supported by the National Natural Science Foundation of China (Grant No. 11271191) and National Basic Research Program of China (Grant No. 2013CB834100).

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Authors and Affiliations

  1. School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Depeng Kong & Ercai Chen

  2. Center of Nonlinear Science, Nanjing University, Nanjing, 210093, People’s Republic of China

    Ercai Chen

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  1. Depeng Kong
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  2. Ercai Chen
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Correspondence to Ercai Chen.

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Kong, D., Chen, E. Slow Entropy for Noncompact Sets and Variational Principle. J Dyn Diff Equat 26, 477–492 (2014). https://doi.org/10.1007/s10884-014-9397-7

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  • DOI: https://doi.org/10.1007/s10884-014-9397-7

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