Topological entropy, upper Carathéodory capacity and fractal dimensions of semigroup actions
Volume 163 / 2021
Abstract
We study dynamical systems given by the action $T: G \times X \to X$ 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 $\{T(g,-)\colon g\in G_1\setminus \{1\}\}$.
