Articles

THE PROXIMAL RELATION, REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS

  • Yuan LIAN ,
  • Xiaojun HUANG ,
  • Zhiqiang LI
Expand
  • 1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China;
    2. Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing University, Chongqing 401331, China
Yuan LIAN,E-mail:20140602035@cqu.edu.cn;Zhiqiang LI,E-mail:zqli@cqu.edu.cn

Received date: 2019-11-06

  Revised date: 2020-08-20

  Online published: 2021-06-07

Supported by

The second author is supported by NSF of China (11671057), NSF of Chongqing (cstc2020jcyj-msxmX0694) and the Fundamental Research Funds for the Central Universities (2018CDQYST0023).

Abstract

In this paper, we study the proximal relation, regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions. A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation, and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action.

Cite this article

Yuan LIAN , Xiaojun HUANG , Zhiqiang LI . THE PROXIMAL RELATION, REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS[J]. Acta mathematica scientia, Series B, 2021 , 41(3) : 729 -752 . DOI: 10.1007/s10473-021-0307-x

References

[1] Auslander J. Mean-L-stable systems. Illinois J Math, 1959, 3:566-579
[2] Auslander J. Minimal Flows and Their Extensions. North-Holland, 1988
[3] Aujogue J B, Barge M, Kellendonk J, Lenz D. Equicontinuous Factors, Proximality and Ellis Semigroup for Delone Sets. Birkhäuser Basel:Mathematics of Aperiodic Order, 2015
[4] Beiglböck M, Bergelson V, Fish A. Sumset phenomenon in countable amenable groups. Adv Math, 2010, 223(2):416-432
[5] Clay J P. Proximity relations in transformation groups. Trans Amer Math Soc, 1963, 108(1):88-96
[6] Denker M, Grillenberger C, Sigmund K. Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics. Vol 527. Berlin-New York:Springer-Verlag, 1976
[7] Downarowicz T, Glasner Eli. Isomorphic extensions and applications. Topol Methods Nonlinear Anal, 2016, 48(1):321-338
[8] Downarowicz T, Huczek D, Zhang G. Tilings of amenable groups. J Reine Angew Math, 2019, 747:277-298
[9] Dooley A H, Zhang G. Local entropy theory of a random dynamical system. Mem Amer Math Soc, 2015, 233(1099)
[10] Downarowicz T, Zhang G. Symbolic extensions of amenable group actions and the comparison property. arXiv preprint arXiv:1901.01457, 2019
[11] Ellis R, Gottschalk W H. Homomorphisms of transformation groups. Trans Amer Math Soc, 1960, 94:258-271
[12] Fomin S. On dynamical systems with a purely point spectrum. Dokl Akad Nauk SSSR, 1951, 77:29-32(in Russian)
[13] Fuhrmann G, Gröger M, Lenz D. The structure of mean equicontinuous group actions. arXiv preprint. https://arxiv.org/pdf/1812.10219
[14] Huang X, Liu J, Zhu C. The Bowen topological entropy of subsets for amenable group actions. J Math Anal Appl, 2019, 472(2):1678-1715
[15] Huang W, Li J, Thouvenot J, et al. Mean equicontinuity, bounded complexity and discrete spectrum. arXiv preprint. https://arxiv.org/pdf/1806.02980
[16] Kerr D, Li H. Ergodic Theory:Independence and Dichotomies. Springer, 2016
[17] Kerr D, Li H. Soficity, amenability, and dynamical entropy. Amer J Math, 2013, 135(3):721-761
[18] Lącka M, Pietrzyk M. Quasi-uniform convergence in dynamical systems generated by an amenable group action. J Lond Math Soc, 2018, 98(3):687-707
[19] Lindenstrauss E. Pointwise theorems for amenable groups. Invent Math, 2001, 146(2):259-295
[20] Li J. Chaos and entropy for interval maps. J Dyn Differ Equ, 2011, 23(2):333-352
[21] Li J, Tu S. On proximality with Banach density one. J Math Anal Appl, 2014, 416(1):36-51
[22] Li J, Tu S, Ye X. Mean equicontinuity and mean sensitivity. Ergodic Theory Dynam Systems, 2015, 35(8):2587-2612
[23] Moothathu T K S. Syndetically proximal pairs. J Math Anal Appl, 2011, 379(2):656-663
[24] Moothathu T K S, Oprocha P. Synetical proximality and scrambled sets. Topol Methods Nonlinear Anal, 2013, 41(2):421-461
[25] Oxtoby J C. Ergodic sets. Bull Amer Math Soc, 1952, 58:116-136
[26] Ollagnier J M. Ergodic Theory and Statistical Mechanics. Springer-Verlag, 1985
[27] Ornstein D S, Weiss B. Entropy and isomorphism theorems for actions of amenable groups. J Analyse Math, 1987, 48(1):1-141
[28] Oprocha P, Zhang G. Topological aspects of dynamics of pairs, tuples and sets. Recent Progress in General Topology Ⅲ. Paris:Atlantis Press, 2014:665-709
[29] Parthasarathy K R. Introduction to probability and measure. London:Macmillan, 1977
[30] Qiu J, Zhao J. A Note on Mean Equicontinuity. J Dyn Differ Equ, 2020, 32:101-116
[31] Rudin W. Functional analysis. McGraw-Hill, Inc, 1991
[32] Scarpellini B. Stability properties of flows with pure point spectrum. J London Math Soc, 1982, 2(3):451-464
[33] Sigmund K. On minimal centers of attraction and generic points. J Reine Angew Math, 1977, 295:72-79
[34] Varadarajan V S. Groups of automorphisms of Borel spaces. Trans Amer Math Soc, 1963, 109:191-220
[35] Walters P. An Introduction to Ergodic Theory. New York:Springer, 1982
[36] Weiss B. Actions of amenable groups. Topics in dynamics and ergodic theory, 226-260, London Math Soc Lecture Note Ser, 310. Cambridge:Cambridge Univ Press, 2003
[37] Zhou Z. Weakly almost periodic point and measure centre. Science in China, Ser A, 1993, 36(2):142-153
[38] Zhu B, Huang X, Lian Y. The systems with almost Banach mean equicontinuity for abelian group actions. arXiv preprint. https://arxiv.org/pdf/1909.00920
Options
Outlines

/