Packing topological entropy for amenable group actions

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...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 43; no. 2; pp. 480 - 514
Main Authors DOU, DOU, ZHENG, DONGMEI, ZHOU, XIAOMIN
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.02.2023
Subjects
Dynamical systems
Entropy
Metric space
Original Article
Topology
packing topological entropy
variational principle
37B40
generic point
28D20
37A15
amenable group
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2021.126