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  • Topological entropy, upper Carathéodory capacity and fractal dimensions of semigroup actions
    Biś, Andrzej ...
    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 ... ▫$T(1,-)=\operatorname{id}_X$▫. 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. [E. Ghys, R. Langevin et P. Walczak, Entropie géométrique des feuilletages, Acta Math. 160 (1988), 105-142] and Hofmann and Stoyanov [K. H. Hofmann and L. Stoyanov, Topological entropy of groups and semigroup actions, Adv. Math. 115 (1995), 54-98]) 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 ▫$\{T(g,-):g \in G_1 \setminus \{1\}\}$▫.
    Vir: Colloquium mathematicum. - ISSN 0010-1354 (Vol. 163, no. 1, 2021, str. 131-151)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2021
    Jezik - angleški
    COBISS.SI-ID - 65377539

    Povezava(-e):

    http://dx.doi.org/10.4064/cm8017-12-2019
    DOI

vir: Colloquium mathematicum. - ISSN 0010-1354 (Vol. 163, no. 1, 2021, str. 131-151)

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