Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-09T18:40:32.525Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite-step nilsystems, independence and complexity

Published online by Cambridge University Press:  09 December 2011

PANDENG DONG ,
SEBASTIÁN DONOSO ,
ALEJANDRO MAASS ,
SONG SHAO  and
XIANGDONG YE
PANDENG DONG
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, PR China Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (email: dopandn@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
SEBASTIÁN DONOSO
Affiliation:
Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile (email: sdonoso@dim.uchile.cl, amaass@dim.uchile.cl)
ALEJANDRO MAASS
Affiliation:
Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile (email: sdonoso@dim.uchile.cl, amaass@dim.uchile.cl)
SONG SHAO
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (email: dopandn@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
XIANGDONG YE
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (email: dopandn@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

An ∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an ∞-step nilsystem if and only if it has no non-trivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without non-trivial pairs with arbitrarily long finite IP-independence sets is an almost one-to-one extension of its maximal ∞-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some ∞-step nilsystem. The question if such a system is uniquely ergodic remains open. In addition, the topological complexity of an ∞-step nilsystem is computed, showing that it is polynomial for each non-trivial open cover.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 118 - 143
Copyright
Copyright © Cambridge University Press 2013

References

[1]Auslander, J.. Minimal Flows and Their Extensions (North-Holland Mathematics Studies, 153). North-Holland, Amsterdam, 1988.Google Scholar
[2]Auslander, L., Green, L. and Hahn, F.. Flows on Homogeneous Spaces (Annals of Mathematics Studies, 53). Princeton University Press, Princeton, NJ, 1963.CrossRefGoogle Scholar
[3]Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160 (2005), 261–303 with an appendix by I.Z. Ruzsa.Google Scholar
[4]Blanchard, F., Host, B. and Maass, A.. Topological complexity. Ergod. Th. & Dynam. Sys. 20 (2000), 641–662.CrossRefGoogle Scholar
[5]Blanchard, F. and Lacroix, Y.. Zero-entropy factors of topological flows. Proc. Amer. Math. Soc. 119 (1993), 985–992.Google Scholar
[6]Furstenberg, H.. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601.Google Scholar
[7]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory (M. B. Porter Lectures). Princeton University Press, Princeton, NJ, 1981.Google Scholar
[8]Glasner, E.. Proximal Flows (Lecture Notes in Mathematics, 517). Springer, New York, 1976.CrossRefGoogle Scholar
[9]Glasner, E.. Topological weak mixing and quasi-Bohr systems. Israel J. Math. 148 (2005), 277–304.Google Scholar
[10]Glasner, E.. The structure of tame minimal dynamical systems. Ergod. Th. & Dynam. Sys. 27 (2007), 1819–1837.Google Scholar
[11]Glasner, E. and Weiss, B.. Quasi-factors of zero-entropy systems. J. Amer. Math. Soc. 8 (1995), 665–686.Google Scholar
[12]Glasner, E. and Ye, X.. Local entropy theory. Ergod. Th. & Dynam. Sys. 29 (2009), 321–356.Google Scholar
[13]Green, B. and Tao, T.. The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. (2) to appear.Google Scholar
[14]Host, B.. Convergence of multiple ergodic averages, arXiv:0606362.Google Scholar
[15]Host, B. and Kra, B.. Nonconventional averages and nilmanifolds. Ann. of Math. (2) 161 (2005), 398–488.Google Scholar
[16]Host, B., Kra, B. and Maass, A.. Nilsequences and a structure theory for topological dynamical systems. Adv. Math. 224 (2010), 103–129.CrossRefGoogle Scholar
[17]Huang, W.. Tame systems and scrambled pairs under an Abelian group action. Ergod. Th. & Dynam. Sys. 26 (2006), 1549–1567.Google Scholar
[18]Huang, W., Li, H. and Ye, X.. Family-independence for topological and measurable dynamics. Trans. Amer. Math. Soc. arXiv:0908.0574 to appear.Google Scholar
[19]Huang, W., Li, H. and Ye, X.. Localization and dynamical Ramsey property. Preprint, 2010.Google Scholar
[20]Huang, W., Li, S., Shao, S. and Ye, X.. Null systems and sequence entropy pairs. Ergod. Th. & Dynam. Sys. 23 (2003), 1505–1523.Google Scholar
[21]Huang, W. and Ye, X.. Topological complexity, return times and weak disjointness. Ergod. Th. & Dynam. Sys. 24 (2004), 825–846.CrossRefGoogle Scholar
[22]Huang, W. and Ye, X.. A local variational relation and applications. Israel J. Math. 151 (2006), 237–280.Google Scholar
[23]Kra, B.. Personal communication.Google Scholar
[24]Kerr, D. and Li, H.. Independence in topological and C *-dynamics. Math. Ann. 338 (2007), 869–926.Google Scholar
[25]Leibman, A.. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25(1) (2005), 201–213.Google Scholar
[26]Lindenstrauss, E.. Measurable distal and topological distal systems. Ergod. Th. & Dynam. Sys. 19(4) (1999), 1063–1076.Google Scholar
[27]Maass, A. and Shao, S.. Sequence entropy in minimal systems. J. Lond. Math. Soc. 76(3) (2007), 702–718.Google Scholar
[28]Pollicott, M.. Fractals and Dimension Theory. Available at http://www.warwick.ac.uk/∼masdbl/preprints.html.Google Scholar
[29]Shao, S. and Ye, X.. Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence, arXiv:1007.0189.Google Scholar
[30]Veech, W. A.. Point-distal flows. Amer. J. Math. 92 (1970), 205–242.CrossRefGoogle Scholar
[31]Veech, W. A.. Topological dynamics. Bull. Amer. Math. Soc. 83 (1977), 775–830.CrossRefGoogle Scholar
[32]Weiss, B.. Multiple recurrence and doubly minimal systems. Contemp. Math. 215 (1998), 189–196.Google Scholar
[33]van der Woude, J.. Topological dynamics. Dissertation, Vrije Universiteit, Amsterdam, 1982. CWI Tract, 22.Google Scholar
[34]Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 53–97.Google Scholar
[35]Zygmund, A.. Trigonometric Series, 2nd edn. University Press, Cambridge, 1959.Google Scholar