Metric Versus Topological Receptive Entropy of Semigroup Actions

Biś, Andrzej; Dikranjan, Dikran; Bruno, Anna Giordano; Stoyanov, Luchezar

doi:10.1007/s12346-021-00485-7

Cited by 6 publications

(2 citation statements)

References 48 publications

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“…This Bridge Theorem for the receptive entropy will be exploited in [BDGS3] to obtain further properties of the receptive topological entropy of semigroup actions via continuous endomorphisms of profinite (necessarily) compact abelian groups.…”

Section: Comparisons Between Metric and Topological Entropymentioning

confidence: 99%

Metric vs topological receptive entropy of semigroup actions

Biś,

Dikranjan,

Bruno

et al. 2021

Preprint

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We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov [HS]. We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin-Katok Formula and the local Variational Principle.

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Section: Comparisons Between Metric and Topological Entropymentioning

confidence: 99%

Metric vs topological receptive entropy of semigroup actions

Biś,

Dikranjan,

Bruno

et al. 2021

Preprint

Self Cite

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“…For further extensions of these classical entropies to the case of actions of sofic groups see the survey paper [36] by Weiss. In a different direction, Bowen's topological entropy was generalized by Ghys, Langevin and Walczak [14] to finitely generated pseudogroups of local homeomorphisms of a compact metric space in the setting of foliations. Independently, a more general notion for locally compact semigroup actions on a compact metric space was introduced by Hofmann and Stoyanov [18] (see also [4] where the term "receptive" was coined for this kind of entropy).…”

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confidence: 99%

Topological entropy, upper Carath

et al. 2021

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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) andHofmann andStoyanov (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}}.

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Hilbert polynomial of length functions

Fornasiero

2024

Annali di Matematica

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Let $$\lambda $$ λ be a general length function for modules over a Noetherian ring R. We use $$\lambda $$ λ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of $$\lambda $$ λ . We show that the leading term $$\mu $$ μ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for R[X]-modules. Similar to algebraic entropy, $$\mu $$ μ in general is not additive for exact sequences of R[X]-modules: we demonstrate how to adapt certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.

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Metric Versus Topological Receptive Entropy of Semigroup Actions

Cited by 6 publications

References 48 publications

Metric vs topological receptive entropy of semigroup actions

Metric vs topological receptive entropy of semigroup actions

Topological entropy, upper Carath

Hilbert polynomial of length functions

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