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

Tom 163 / 2021

Andrzej Biś, Dikran Dikranjan, Anna Giordano Bruno, Luchezar Stoyanov Colloquium Mathematicum 163 (2021), 131-151 MSC: 37B40, 37A35, 28D20, 54H15, 28A75. DOI: 10.4064/cm8017-12-2019 Opublikowany online: 15 June 2020

Streszczenie

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 $\{T(g,-)\colon g\in G_1\setminus \{1\}\}$.

Autorzy

  • Andrzej BiśFaculty of Mathematics and Computer Science
    University of Łódź
    Banacha 22
    90-235 Łódź, Poland
    e-mail
  • Dikran DikranjanDepartment of Mathematics,
    Computer Science and Physics
    University of Udine
    Via delle Scienze 206
    33100 Udine, Italy
    e-mail
  • Anna Giordano BrunoDepartment of Mathematics,
    Computer Science and Physics
    University of Udine
    Via delle Scienze 206
    33100 Udine, Italy
    e-mail
  • Luchezar StoyanovMathematics and Statistics
    Faculty of Engineering and Mathematical Sciences
    The University of Western Australia
    35 Stirling Highway
    6009 Perth, Australia
    e-mail

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