The authors’ aim is to introduce a general notion of topological entropy of a uniform action of a locally compact semigroup on a metric space and derive its basic properties; particular emphasis is given to the case of locally compact groups. The notion of entropy introduced in this paper is a generalization (both in fact and in flavor) of the usual notion of topological entropy for a single mapping, but is distinct from previous notions of topological entropy for amenable group actions. It is dependent on the choice of a sequence \(N_n= (N_1)^n\) of compact sets containing the identity of the acting semigroup. It is shown to be an upper bound for each individual entropy of each member of the acting group or semigroup.
The authors develop general features of this notion of entropy, relate the entropy of an action to that of subactions and quotient actions, and tie it to measure theoretic entropy in the case a homogeneous measure exists. The case of linear actions of locally compact groups of finite dimensional spaces is given special attention. It turns out that every locally compact connected group admitting a finite-dimensional representation with nonzero entropy has exponential (as opposed to polynomial) growth.