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 definition, each h-expansive action of a sofic group is asymptotically h-expansive. We show that each expansive action of a sofic group is h-expansive, and, for any given asymptotically h-expansive action of a sofic group, 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 first introduced and studied by Misiurewicz for Z-actions using the language of tail 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.