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.