Topological entropy, upper Caratheodory capacity and fractal dimensions of semigroup actions

We study dynamical systems given by the action T : G x 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 Caratheodory 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 delta-measure entropy, for delta is an element of (0, 1).In the case when T(g, -) is a locally expanding selfmap of X for every g is an element of G {1}, we show that the receptive topological entropy of G(1) dominates the Hausdorff dimension of X modulo a factor log lambda determined by the expanding coefficients of the elements of {T(g, -) : g is an element of G(1) {1}}.