TY - JOUR

T1 - Topological entropy, upper carathÉodory capacity and fractal dimensions of semigroup actions

AU - Biś, Andrzej

AU - Dikranjan, Dikran

AU - Bruno, Anna Giordano

AU - Stoyanov, Luchezar

PY - 2021

Y1 - 2021

N2 - 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, −) = idX. 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}}.

AB - 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, −) = idX. 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}}.

KW - Carathéodory structure

KW - finitely generated semigroup action

KW - Fractal dimension

KW - Hausdorff dimension

KW - Measure entropy

KW - Receptive entropy

KW - Topological entropy

KW - Upper capacity

UR - http://www.scopus.com/inward/record.url?scp=85104116297&partnerID=8YFLogxK

U2 - 10.4064/CM8017-12-2019

DO - 10.4064/CM8017-12-2019

M3 - Article

AN - SCOPUS:85104116297

VL - 163

SP - 131

EP - 151

JO - Colloquium Mathematicum

JF - Colloquium Mathematicum

SN - 0010-1354

IS - 1

ER -