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MiS Preprint

Weak Expansiveness for Actions of Sofic Groups

Nhân Phú Chung and Guohua Zhang


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 $\mathbb{Z}$-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.

MSC Codes:
37B05, 37C85, 54H15
sofic groups, entropy, asymptotically h-expansiveness

Related publications

2015 Repository Open Access
Nhan Phú Chung and Guohua Zhang

Weak expansiveness for actions of sofic groups

In: Journal of functional analysis, 268 (2015) 11, pp. 3534-3565