The Scaling Entropy of a Generic Action

Veprev, Georgii

doi:10.1007/s10958-022-05773-4

Cited by 5 publications

(1 citation statement)

References 7 publications

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“…It turns out that the scaling entropy of a generic automorphism T ∈ Aut(X, µ) is not comparable to an arbitrary fixed function (except for the trivial extreme cases of bounded and linear growth). The following theorem was proved in [49].…”

Section: Generic Scaling Entropymentioning

confidence: 98%

Dynamics of metrics in measure spaces and scaling entropy

Vershik,

Veprev,

Zatitskii

2023

Russian Math. Surveys

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This survey is dedicated to a new direction in the theory of dynamical systems, the dynamics of metrics in measure spaces and new (catalytic) invariants of transformations with invariant measure. A space equipped with a measure and a metric which are naturally consistent with each other (a metric triple, or an mm-space) defines automatically the notion of its entropy class, thus allowing one to construct a theory of scaling entropy for dynamical systems with invariant measure, which is different and more general in comparison to the Shannon-Kolmogorov theory. This possibility was hinted at by Shannon himself, but the hint went unnoticed. The classification of metric triples in terms of matrix distributions presented in this paper was proposed by Gromov and Vershik. We describe some corollaries obtained by applying this theory. Bibliography: 88 titles.

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Section: Generic Scaling Entropymentioning

confidence: 98%

Dynamics of metrics in measure spaces and scaling entropy

Vershik,

Veprev,

Zatitskii

2023

Russian Math. Surveys

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Non-existence of a universal zero-entropy system via generic actions of almost complete growth

VEPREV

2024

Ergod. Th. Dynam. Sys.

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We prove that a generic probability measure-preserving (p.m.p.) action of a countable amenable group G has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. G-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.

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Dynamics of metrics in measure spaces and scaling entropy

Вершик¹,

Veprev

Zatitskii

2023

Uspekhi Mat. Nauk

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Настоящий обзор посвящен новому направлению в теории динамических систем: динамике метрик в пространствах с мерой и новым (каталитическим) инвариантам преобразований с инвариантной мерой. Пространство с согласованными мерой и метрикой (метрические тройки или mm-пространства) автоматически определяет понятие своего энтропийного класса, позволяет построить иную и более общую по сравнению с теорией Шеннона-Колмогорова теорию энтропии динамических систем с инвариантной мерой. Незамеченный ранее намек на такую возможность высказывал еще К. Шеннон. Приводимая в статье классификация метрических троек с помощью матричных распределений принадлежит М. Громову и А. М. Вершику. Приводятся некоторые следствия и применения излагаемой теории. Библиография: 88 названий.

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The Scaling Entropy of a Generic Action

Cited by 5 publications

References 7 publications

Dynamics of metrics in measure spaces and scaling entropy

Dynamics of metrics in measure spaces and scaling entropy

Non-existence of a universal zero-entropy system via generic actions of almost complete growth

Dynamics of metrics in measure spaces and scaling entropy

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