We will study dynamical systems given by the action of finitely generated semigroup $G$ on compact metric space $X$ by continuous selfmaps. The main object in this talk will be relating several entropy-like quantities for semigroup actions and obtaining partial variational principles. For a finite generating set $G_1$ of $G$, the receptive topological entropy of $G_1$ (in the sense of Ghys-Langevin-Walczak (1988) and Hofmann-Stoyanov (1995)) is shown to coincide with the limit of upper capacities of dynamically defined Carathéodory structures on $X$ depending on $G_1.$ Similar result holds for the classical topological entropy of amenable semigroup.
Moreover, the receptive topological entropy and the topological entropy of $G_1$ are lower bounded by respective generalizations of Katok's $\delta-$measure entropy, for $\delta \in (0,1).$ In the case when the semigroup $G$ acting on a compact metric space $X$, is $\lambda-$locally expanding, its receptive topological entropy dominates the Hausdorff dimension of X modulo $\log(\lambda).$
The talk is based on joint works with Dikran Dikranjan, Anna Giordano Bruno and Luchezar Stoyanov.