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

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 G₁ of G containing 1, the receptive topological entropy of G₁ (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 G₁, 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 G₁ 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 G₁ dominates the Hausdorff dimension of X modulo a factor log λ determined by the expanding coefficients of the elements of {T (g, −): g ∈ G₁ \ {1}}.