February  2015, 35(2): 725-740. doi: 10.3934/dcds.2015.35.725

Localization of mixing property via Furstenberg families

1. 

Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China

Received  November 2013 Revised  April 2014 Published  September 2014

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.
Citation: Jian Li. Localization of mixing property via Furstenberg families. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 725-740. doi: 10.3934/dcds.2015.35.725
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.

show all references

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.

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