Localization of mixing property via Furstenberg families

Jian Li

doi:10.3934/dcds.2015.35.725

Article Contents

2015, Volume 35, Issue 2: 725-740. Doi: 10.3934/dcds.2015.35.725

This issue Previous Article Unbounded perturbations of the generator domain Next Article Chain transitive induced interval maps on continua

Localization of mixing property via Furstenberg families

Jian Li^1,

1.
Department of Mathematics, Shantou University, Shantou, Guangdong 515063

Received: October 31, 2013

Revised: March 31, 2014

Published: January 2015

Abstract Related Papers Cited by

Abstract

This paper is devoted to studying the localization of mixing property via Furstenberg families. It is shown that there exists some $\mathcal{F}_{pubd}$-mixing set in every dynamical system with positive entropy, and some $\mathcal{F}_{ps}$-mixing set in every non-PI minimal system.

Keywords:

Mathematics Subject Classification: 54H20, 37B05, 37B40.

Citation:

Related Papers

Cited by

References

[1]	E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, The University Series in Mathematics, Plenum Press, New York, 1997.doi: 10.1007/978-1-4757-2668-8.
[2]	E. Akin, Lectures on Cantor and Mycielski sets for dynamical systems, in Chapel Hill Ergodic Theory Workshops, Contemp. Math., 356, Amer. Math. Soc., Providence, RI, 2004, 21-79.doi: 10.1090/conm/356/06496.
[3]	E. Akin, E. Glasner, W. Huang, S. Shao and X. Ye, Sufficient conditions under which a transitive system is chaotic, Ergod. Th. and Dynam. Sys., 30 (2010), 1277-1310.doi: 10.1017/S0143385709000753.
[4]	F. Blanchard, Fully positive topological entropy and topological mixing, Symbolic Dynamics and its Applications (New Haven, CT, 1991), Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992, 95-105.doi: 10.1090/conm/135/1185082.
[5]	F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France, 121 (1993), 465-478.
[6]	F. Blanchard, B. Host, A. Maass, S. Martinez and D. Rudolph, Entropy pairs for a measure, Ergod. Theory Dynam. Syst., 15 (1995), 621-632.doi: 10.1017/S0143385700008579.
[7]	F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity, Discrete Contin. Dyn. Syst., 20 (2008), 275-311.
[8]	D. Dou, X. Ye and G. Zhang, Entropy sequence and maximal entropy sets, Nonlinearity, 19 (2006), 53-74.doi: 10.1088/0951-7715/19/1/004.
[9]	R. Ellis, Extending continuous functions on zero-dimensional spaces, Math. Ann., 186 (1970), 114-122.doi: 10.1007/BF01350686.
[10]	R. Ellis, S. Glasner and L. Shapiro, Proximal-Isometric Flows, Advances in Math., 17 (1975), 213-260.doi: 10.1016/0001-8708(75)90093-6.
[11]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.
[12]	W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Collooquium Publications, Vol. 36, Providence, R.I., 1955.
[13]	E. Glasner, A simple characterization of the set of $\mu$-entropy pairs and applications, Israel J. Math., 102 (1997), 13-27.doi: 10.1007/BF02773793.
[14]	E. Glasner, Topological weak mixing and quasi-Bohr systems, Israel J. Math., 148 (2005), 277-304.doi: 10.1007/BF02775440.
[15]	E. Glasner and X. Ye, Local entropy theory, Ergodic Theory and Dynam. Systems, 29 (2009), 321-356.doi: 10.1017/S0143385708080309.
[16]	E. Glasner, Classifying dynamical systems by their recurrence properties, Topol. Methods Nonlinear Anal., 24 (2004), 21-40.
[17]	W. Huang, S. Shao and X. Ye, Mixing and proximal cells along a sequences, Nonlinearity, 17 (2004), 1245-1260.doi: 10.1088/0951-7715/17/4/006.
[18]	W. Huang and X. Ye, Dynamical systems disjoint from and minimal system, Tran. Amer. Math. Soc., 357 (2005), 669-694.doi: 10.1090/S0002-9947-04-03540-8.
[19]	W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergod. Thero. Dyn. Syst., 24 (2004), 825-846.doi: 10.1017/S0143385703000543.
[20]	W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.doi: 10.1007/BF02777364.
[21]	A. Illanes and S. Nadler, Hyperspaces, Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, Inc., New York, 1999.
[22]	J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231.doi: 10.1016/j.topol.2011.07.013.
[23]	J. Li, P. Oprocha and G. Zhang, On recurrence over subsets and weak mixing, preprint, 2013.
[24]	J. Mycielski, Independent sets in topological algebras, Fund. Math., 55 (1964), 139-147.
[25]	P. Oprocha, Coherent lists and chaotic sets, Discrete Continuous Dynam. Systems, 31 (2011), 797-825.doi: 10.3934/dcds.2011.31.797.
[26]	P. Oprocha and G. Zhang, On local aspects of topological weak mixing in dimension one and beyond, Studia Math., 202 (2011), 261-288.doi: 10.4064/sm202-3-4.
[27]	P. Oprocha and G. Zhang, On sets with recurrence properties, their topological structure and entropy, Top. App., 159 (2012), 1767-1777.doi: 10.1016/j.topol.2011.04.020.
[28]	P. Oprocha and G. Zhang, On weak product recurrence and synchroniztion of return times, Adv. Math., 244 (2013), 395-412.doi: 10.1016/j.aim.2013.05.006.
[29]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
[30]	J. Xiong and Z. Yang, Chaos caused by a toplogical mixing map, in Dynamical Systems and Related Topics (Nagoya, 1990), Adv. Ser. Dynam. Systems, 9, World Sci. Publ., River Edge, NJ, 1991, 550-572.