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

Part of: Ergodic theory Measure-theoretic ergodic theory

Published online by Cambridge University Press:  12 April 2024

GEORGII VEPREV
GEORGII VEPREV*
Affiliation:
Leonhard Euler International Mathematical Institute in St. Petersburg, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia
*
e-mail: georgii.veprev@gmail.com
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Abstract

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.

Keywords

generic actionuniversal systemscaling entropyamenable group actionscaling entropy growth gap

MSC classification

Primary: 37A15: General groups of measure-preserving transformations 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 28D15: General groups of measure-preserving transformations 28D20: Entropy and other invariants
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 15
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Adams, T.. Genericity and rigidity for slow entropy transformations. New York J. Math. 27 (2021), 393–416.Google Scholar
Downarowicz, T. and Serafin, J.. Universal systems for entropy intervals. J. Dynam. Differential Equations 29 (2017), 1411–1422.CrossRefGoogle Scholar
Ferenczi, S.. Measure-theoretic complexity of ergodic systems. Israel J. Math. 100 (1997), 187–207.CrossRefGoogle Scholar
Foreman, M. and Weiss, B.. An anti-classification theorem for ergodic measure preserving transformations. J. Eur. Math. Soc. (JEMS) 6 (2004), 277–292.CrossRefGoogle Scholar
Helfgott, H. A.. Growth and generation in SL2(Z/pZ). Ann. of Math. (2) 167(2) (2008), 601–623.CrossRefGoogle Scholar
Jordan, H. E.. Group-characters of various types of linear groups. Amer. J. Math. 29 (1907), 387–405.CrossRefGoogle Scholar
Kanigowski, A., Katok, A. and Wei, D.. Survey on entropy-type invariants of sub-exponential growth in dynamical systems. A Vision for Dynamics in the 21st Century: The Legacy of Anatole Katok. Cambridge University Press, Cambridge, 2024, pp 232–289.CrossRefGoogle Scholar
Katok, A. and Thouvenot, J.-P.. Slow entropy type invariants and smooth realization of commuting measure-preserving transformations. Ann. Inst. Henri Poincaré D 33 (1997), 323–338.CrossRefGoogle Scholar
Kechris, A.. Global Aspects of Ergodic Group Actions (Mathematical Surveys and Monographs, 160). American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Ergodic Theory: Independence and Dichotomies. Springer, Cham, 2017.Google Scholar
Krieger, W.. On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc. 149 (1970), 453–464.CrossRefGoogle Scholar
Kushnirenko, A. G.. On metric invariants of the type of entropy. Uspekhi Mat. Nauk 22(5) (1967), 57–65.Google Scholar
Malcev, A. I.. On isomorphic matrix representations of infinite groups. Rec. Math. [Mat. Sb.] (N.S.) 8(50) (1940), 3, 405–422.Google Scholar
Petrov, F. V., Vershik, A. M. and Zatitskiy, P. B.. Geometry and dynamics of admissible metrics in measure spaces. Cent. Eur. J. Math. 11(3) (2013), 379–400.Google Scholar
Petrov, F. V. and Zatitskiy, P. B.. On the subadditivity of a scaling entropy sequence. J. Math. Sci. (N.Y.) 215(6) (2016), 734–737.Google Scholar
Pyber, L. and Szabó, E.. Growth in finite simple groups of Lie type. J. Amer. Math. Soc. 29(1) (2016), 95–146.CrossRefGoogle Scholar
Rokhlin, V. A.. Lectures on the entropy theory of measure-preserving transformations. Russian Math. Surveys 22(5) (1967), 1–52.CrossRefGoogle Scholar
Ryzhikov, V. V.. Compact families and typical entropy invariants of measure-preserving actions. Trans. Moscow Math. Soc. 82 (2021), 117–123.CrossRefGoogle Scholar
Schur, I.. Unterschungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 132 (1906–07), 85–137.Google Scholar
Serafin, J.. Non-existence of a universal zero-entropy system. Israel J. Math. 194(1) (2013), 349–358.CrossRefGoogle Scholar
Shilon, O. and Weiss, B.. Universal minimal topological dynamical systems. Israel J. Math. 160 (2007), 119–141.CrossRefGoogle Scholar
Veprev, G.. Scaling entropy of unstable systems. J. Math. Sci. (N.Y.) 254(2) (2021), 109–119.CrossRefGoogle Scholar
Veprev, G.. The scaling entropy of a generic action. J. Math. Sci. (N.Y.) 261 (2022), 595–600.CrossRefGoogle Scholar
Veprev, G.. Non-existence of a universal zero entropy system for non-periodic amenable group actions. Israel J. Math. 253 (2023), 715–743.CrossRefGoogle Scholar
Vershik, A. M.. Information, entropy, dynamics. Mathematics of the 20th Century: A View from Petersburg. Ed. A. M. Vershik. MCCME, Moscow, 2010, pp. 47–76 (in Russian).Google Scholar
Vershik, A. M.. Dynamics of metrics in measure spaces and their asymptotic invariants. Markov Process. Related Fields 16(1) (2010), 169–185.Google Scholar
Vershik, A. M.. Scaling entropy and automorphisms with pure point spectrum. St. Petersburg Math. J. 23(1) (2012), 75–91.CrossRefGoogle Scholar
Vershik, A. M. and Zatitskiy, P. B.. Universal adic approximation, invariant measures and scaled entropy. Izv. Math. 81(4) (2017), 734–770.CrossRefGoogle Scholar
Vershik, A. M. and Zatitskiy, P. B.. Combinatorial invariants of metric filtrations and automorphisms; the universal adic graph. Funct. Anal. Appl. 52 (2018), 258–269.CrossRefGoogle Scholar
Weiss, B.. Countable generators in dynamics-universal minimal models. Contemp. Math. 94 (1989), 321–326.CrossRefGoogle Scholar
Yu, T., Zhang, G. and Zhang, R.. Discrete spectrum for amenable group actions. Discrete Contin. Dyn. Syst. 41(12) (2021), 5871.CrossRefGoogle Scholar
Zatitskiy, P. B.. Scaling entropy sequence: invariance and examples. J. Math. Sci. (N.Y.) 209(6) (2015), 890–909.CrossRefGoogle Scholar
Zatitskiy, P. B.. On the possible growth rate of a scaling entropy sequence. J. Math. Sci. (N.Y.) 215(6) (2016), 715–733.CrossRefGoogle Scholar