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Packing topological entropy for amenable group actions

Part of: Ergodic theory Topological dynamics Measure-theoretic ergodic theory

Published online by Cambridge University Press:  20 October 2021

DOU DOU ,
DONGMEI ZHENG  and
XIAOMIN ZHOU Open the ORCID record for XIAOMIN ZHOU [Opens in a new window]
DOU DOU*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, P. R. China
DONGMEI ZHENG
Affiliation:
School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, Jiangsu 211816, P. R. China (e-mail: dongmzheng@163.com)
XIAOMIN ZHOU
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P. R. China (e-mail: zxm12@mail.ustc.edu.cn)
*
e-mail: doumath@163.com
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Abstract

Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$, where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

Keywords

packing topological entropyamenable groupvariational principlegeneric point

MSC classification

Primary: 37B40: Topological entropy 28D20: Entropy and other invariants 37A15: General groups of measure-preserving transformations

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 2 , February 2023 , pp. 480 - 514
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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