Hostname: page-component-77f85d65b8-8v9h9 Total loading time: 0 Render date: 2026-04-21T08:15:28.348Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial independence and naive entropy

Part of: Topological dynamics Extremal combinatorics Ergodic theory

Published online by Cambridge University Press:  15 May 2020

HANFENG LI  and
ZHEN RONG
HANFENG LI
Affiliation:
Center of Mathematics, Chongqing University, Chongqing 401331, China Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, USA email hfli@math.buffalo.edu
ZHEN RONG
Affiliation:
College of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010000, China email rongzhen@cqu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li–Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.

Keywords

naive entropycombinatorial independenceLi–Yorke chaosdistal actiontame action

MSC classification

Primary: 37B40: Topological entropy 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 05D10: Ramsey theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2136 - 2147
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aujogue, J.-B.. Ellis enveloping semigroup for almost canonical model sets of an Euclidean space. Algebr. Geom. Topol. 15(4) (2015), 2195–2237.CrossRefGoogle Scholar
Auslander, J.. Minimal Flows and their Extensions (North-Holland Mathematics Studies, 153. Notas de Matemática [Mathematical Notes], 122) . North-Holland, Amsterdam, 1988.Google Scholar
Blanchard, F.. Fully positive topological entropy and topological mixing. Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135) . American Mathematical Society, Providence, RI, 1992, pp. 95–105.CrossRefGoogle Scholar
Blanchard, F.. A disjointness theorem involving topological entropy. Bull. Soc. Math. France 121(4) (1993), 465–478.CrossRefGoogle Scholar
Blanchard, F., Glasner, E. and Host, B.. A variation on the variational principle and applications to entropy pairs. Ergod. Th. & Dynam. Sys. 17(1) (1997), 29–43.CrossRefGoogle Scholar
Blanchard, F., Glasner, E., Kolyada, S. and Maass, A.. On Li–Yorke pairs. J. Reine Angew. Math. 547 (2002), 51–68.Google Scholar
Bowen, L.. Examples in the entropy theory of countable group actions. Ergod. Th. & Dynam. Sys. https://doi.org/10.1017/etds.2019.18. Published online 25 March 2019.CrossRefGoogle Scholar
Burton, P.. Naive entropy of dynamical systems. Israel J. Math. 219(2) (2017), 637–659.CrossRefGoogle Scholar
Chernikov, A. and Simon, P.. Definably amenable NIP groups. J. Amer. Math. Soc. 31(3) (2018), 609–641.CrossRefGoogle Scholar
Chung, N. and Li, H.. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. Math. 199(3) (2015), 805–858.CrossRefGoogle Scholar
Downarowicz, T., Frej, B. and Romagnoli, P.-P.. Shearer’s inequality and infimum rule for Shannon entropy and topological entropy. Dynamics and Numbers (Contemporary Mathematics, 669) . American Mathematical Society, Providence, RI, 2016, pp. 63–75.CrossRefGoogle Scholar
Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101) . American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Glasner, E.. On tame enveloping semigroups. Colloq. Math. 105(2) (2006), 283–295.CrossRefGoogle Scholar
Glasner, E.. The structure of tame minimal dynamical systems. Ergod. Th. & Dynam. Sys. 27(6) (2007), 1819–1837.CrossRefGoogle Scholar
Glasner, E.. The structure of tame minimal dynamical systems for general groups. Invent. Math. 211(1) (2018), 213–244.CrossRefGoogle Scholar
Glasner, E. and Megrelishvili, M.. Hereditarily non-sensitive dynamical systems and linear representations. Colloq. Math. 104(2) (2006), 223–283.CrossRefGoogle Scholar
Glasner, E. and Megrelishvili, M.. Representations of dynamical systems on Banach spaces not containing ℓ 1 . Trans. Amer. Math. Soc. 364(12) (2012), 6395–6424.CrossRefGoogle Scholar
Glasner, E. and Megrelishvili, M.. Representations of dynamical systems on Banach spaces. Recent Progress in General Topology. III. Atlantis Press, Paris, 2014, pp. 399–470.CrossRefGoogle Scholar
Glasner, E. and Megrelishvili, M.. Eventual nonsensitivity and tame dynamical systems. Preprint, 2014,arXiv:1405.2588.Google Scholar
Glasner, E. and Megrelishvili, M.. Circularly ordered dynamical systems. Monatsh. Math. 185(3) (2018), 415–441.CrossRefGoogle Scholar
Glasner, E. and Megrelishvili, M.. More on tame dynamical systems. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213) . Springer, Cham, 2018, pp. 351–392.CrossRefGoogle Scholar
Glasner, E., Megrelishvili, M. and Uspenskij, V. V.. On metrizable enveloping semigroups. Israel J. Math. 164 (2008), 317–332.CrossRefGoogle Scholar
Glasner, E. and Ye, X.. Local entropy theory. Ergod. Th. & Dynam. Sys. 29(2) (2009), 321–356.CrossRefGoogle Scholar
Huang, W.. Tame systems and scrambled pairs under an Abelian group action. Ergod. Th. & Dynam. Sys. 26(5) (2006), 1549–1567.CrossRefGoogle Scholar
Huang, W., Li, H. and Ye, X.. Family independence for topological and measurable dynamics. Trans. Amer. Math. Soc. 364(10) (2012), 5209–5242.Google Scholar
Huang, W., Maass, A., Romagnoli, P. P. and Ye, X.. Entropy pairs and a local Abramov formula for a measure theoretical entropy of open covers. Ergod. Th. & Dynam. Sys. 24(4) (2004), 1127–1153.CrossRefGoogle Scholar
Huang, W. and Ye, X.. A local variational relation and applications. Israel J. Math. 151 (2006), 237–279.CrossRefGoogle Scholar
Huang, W. and Ye, X.. Combinatorial lemmas and applications to dynamics. Adv. Math. 220(6) (2009), 1689–1716.Google Scholar
Huang, W., Ye, X. and Zhang, G.. Local entropy theory for a countable discrete amenable group action. J. Funct. Anal. 261(4) (2011), 1028–1082.CrossRefGoogle Scholar
Ibarlucía, T.. The dynamical hierarchy for Roelcke precompact Polish groups. Israel J. Math. 215(2) (2016), 965–1009.CrossRefGoogle Scholar
Karpovsky, M. G. and Milman, V. D.. Coordinate density of sets of vectors. Discrete Math. 24(2) (1978), 177–184.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Independence in topological and C ∗ -dynamics. Math. Ann. 338(4) (2007), 869–926.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Combinatorial independence in measurable dynamics. J. Funct. Anal. 256(5) (2009), 1341–1386.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Combinatorial independence and sofic entropy. Commun. Math. Stat. 1(2) (2013), 213–257.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Ergodic Theory: Independence and Dichotomies (Springer Monographs in Mathematics) . Springer, Cham, 2016.CrossRefGoogle Scholar
Köhler, A.. Enveloping semigroups for flows. Proc. R. Irish Acad. Sect. A 95(2) (1995), 179–191.Google Scholar
Li, H.. Compact group automorphisms, addition formulas and Fuglede–Kadison determinants. Ann. of Math. (2) 176(1) (2012), 303–347.CrossRefGoogle Scholar
Li, H. and Rong, Z.. Null actions and RIM non-open extensions of strongly proximal actions. Israel J. Math. 235(1) (2020), 139–168.CrossRefGoogle Scholar
Li, T. Y. and Yorke, J. A.. Period three implies chaos. Amer. Math. Monthly 82(10) (1975), 985–992.CrossRefGoogle Scholar
Lind, D. and Schmidt, K.. A survey of algebraic actions of the discrete Heisenberg group. Uspekhi Mat. Nauk 70(4(424)) (2015), 77–142 (in Russian); Engl. transl. Russian Math. Surveys 70(4) (2015), 657–714.Google Scholar
Parry, W.. Zero entropy of distal and related transformations. Topological Dynamics (Symposium, Colorado State University, Fort Collins, CO, 1967). Benjamin, New York, 1968, pp. 383–389.Google Scholar
Passman, D. S.. The Algebraic Structure of Group Rings (Pure and Applied Mathematics) . Wiley-Interscience [John Wiley], New York, 1977.Google Scholar
Romanov, A. V.. Ergodic properties of discrete dynamical systems and enveloping semigroups. Ergod. Th. & Dynam. Sys. 36(1) (2016), 198–214.CrossRefGoogle Scholar
Rosenthal, H. P.. A characterization of Banach spaces containing ℓ 1 . Proc. Natl. Acad. Sci. USA 71 (1974), 2411–2413.CrossRefGoogle Scholar
Sauer, N.. On the density of families of sets. J. Combin. Theory Ser. A 13 (1972), 145–147.CrossRefGoogle Scholar
Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128) . Birkhäuser, Basel, 1995.Google Scholar
Shelah, S.. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. Math. 41 (1972), 247–261.CrossRefGoogle Scholar