Weak expansiveness for actions of sofic groups

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.Comment: to appear in Journal of Functional Analysi

Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 \cite{B}. In this paper we consider the Bowen's entropy for amenable group action dynamical systems and show that under the tempered condition, the Bowen entropy of the whole compact space for a given F{\o}lner sequence equals to the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin-Katok's local entropy formula for dynamical systems with amenable group actions.Comment: 13 page

Local entropy theory for a countable discrete amenable group action

In the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action. For such an action, we introduce entropy tuples in both topological and measure-theoretic settings and build the variational relation between these two kinds of entropy tuples by establishing a local variational principle for a given finite open cover. Moreover, based the idea of topological entropy pairs, we introduce and study two special classes of such an action: uniformly positive entropy and completely positive entropy. Note that in the building of the local variational principle, following Romagnoli's ideas two kinds of measure-theoretic entropy are introduced for finite Borel covers. These two kinds of entropy turn out to be the same, where Danilenko's orbital approach becomes an inevitable tool