We study a topological entropy of a finitely generated semigroup G of continuous selfmaps defined on a compact metric space X. The topological entropy coincides with the limit of upper capacities of dynamically defined Caratheodory structures on X depending on G.

The topological entropy of G is lower estimated by a generalization of Katok's $\delta-$measure entropy, for any $\delta \in (0,1).$ For a semigroup G of $\lambda-$locally expanding selfmaps on X, the topological entropy of G dominates the Hausdorff dimension of X multiplied by $log(\lambda)$. The talk is based on a joint paper with D. Dikranjan, A. Giordano Bruno and L. Stoyanov.