Weak Expansiveness for Actions of Sofic Groups

Nhân Phú Chung and Guohua Zhang

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Submission date: 28. Feb. 2014
Pages: 35
published in: Journal of functional analysis, 268 (2015) 11, p. 3534-3565 
DOI number (of the published article): 10.1016/j.jfa.2014.12.013
Bibtex
MSC-Numbers: 37B05, 37C85, 54H15
Keywords and phrases: sofic groups, entropy, asymptotically h-expansiveness
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Abstract:
In this paper, we shall introduce h-expansiveness and asymptotical h-expansiveness for actions of sofic groups. By the definitions, each h-expansive action of sofic groups is asymptotically h-expansive. We show that each expansive action of sofic groups is h-expansive, and, for any given asymptotically h-expansive action of sofic groups, the entropy function (with respect to measures) is upper semi-continuous and hence the system admits a measure with maximal entropy. Observe that asymptotically h-expansive property was firstly introduced and studied by Misiurewicz for ℤ-actions using the language of topological conditional entropy. And thus in the remaining part of the paper, we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of Misiurewicz’s ideas when the group is amenable. It turns out that these two definitions are equivalent in this setting.