Topological entropy, upper Carathéodory capacity and fractal dimensions of semigroup actions
Volume 163 / 2021
Abstract
We study dynamical systems given by the action of a finitely generated semigroup G with identity 1 on a compact metric space X by continuous selfmaps and with T(1,-)={\rm 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 \delta -measure entropy, for \delta \in (0,1).
In the case when T(g,-) is a locally expanding selfmap of X for every g\in G\setminus \{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,-)\colon g\in G_1\setminus \{1\}\}.
