Extensibility and denseness of periodic semigroup actions

We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup.