Topological entropy, upper Carath

. We study dynamical systems given by the action T : G × X → X of a finitely generated semigroup G with identity 1 on a compact metric space X by continuous selfmaps and with T (1 , − ) = id X . For any finite generating set G 1 of G containing 1 , the receptive topological entropy of G 1 (in the sense of Ghys et al. (1988) and Hofmann and Stoyanov (1995)) is shown to coincide with the limit of upper capacities of dynamically defined Carathéodory structures on X depending on G 1 , and a similar result holds true for the classical topological entropy when G is amenable. Moreover, the receptive topological entropy and the topological entropy of G 1 are lower bounded by respective generalizations of Katok’s δ -measure entropy, for δ ∈ (0 , 1) . In the case when T ( g, − ) is a locally expanding selfmap of X for every g ∈ G \ { 1 } , we show that the receptive topological entropy of G 1 dominates the Hausdorff dimension of X modulo a factor log λ determined by the expanding coefficients of the elements of { T ( g, − ): g ∈ G 1 \ { 1 }} .