Weak Expansiveness for Actions of Sofic Groups
Nhân Phú Chung and Guohua Zhang
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Submission date: 28. Feb. 2014
MSC-Numbers: 37B05, 37C85, 54H15
Keywords and phrases: sofic groups, entropy, asymptotically h-expansiveness
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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.