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 & Continuous Dynamical Systems - A, 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, (1997).  doi: 10.1007/978-1-4757-2668-8.  Google Scholar

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E. Akin, Lectures on Cantor and Mycielski sets for dynamical systems,, in Chapel Hill Ergodic Theory Workshops, (2004), 21.  doi: 10.1090/conm/356/06496.  Google Scholar

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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.  doi: 10.1017/S0143385709000753.  Google Scholar

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F. Blanchard, Fully positive topological entropy and topological mixing,, Symbolic Dynamics and its Applications (New Haven, (1991), 95.  doi: 10.1090/conm/135/1185082.  Google Scholar

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F. Blanchard, A disjointness theorem involving topological entropy,, Bull. Soc. Math. France, 121 (1993), 465.   Google Scholar

[6]

F. Blanchard, B. Host, A. Maass, S. Martinez and D. Rudolph, Entropy pairs for a measure,, Ergod. Theory Dynam. Syst., 15 (1995), 621.  doi: 10.1017/S0143385700008579.  Google Scholar

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F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity,, Discrete Contin. Dyn. Syst., 20 (2008), 275.   Google Scholar

[8]

D. Dou, X. Ye and G. Zhang, Entropy sequence and maximal entropy sets,, Nonlinearity, 19 (2006), 53.  doi: 10.1088/0951-7715/19/1/004.  Google Scholar

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R. Ellis, Extending continuous functions on zero-dimensional spaces,, Math. Ann., 186 (1970), 114.  doi: 10.1007/BF01350686.  Google Scholar

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R. Ellis, S. Glasner and L. Shapiro, Proximal-Isometric Flows,, Advances in Math., 17 (1975), 213.  doi: 10.1016/0001-8708(75)90093-6.  Google Scholar

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H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, M. B. Porter Lectures, (1981).   Google Scholar

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W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Collooquium Publications, (1955).   Google Scholar

[13]

E. Glasner, A simple characterization of the set of $\mu$-entropy pairs and applications,, Israel J. Math., 102 (1997), 13.  doi: 10.1007/BF02773793.  Google Scholar

[14]

E. Glasner, Topological weak mixing and quasi-Bohr systems,, Israel J. Math., 148 (2005), 277.  doi: 10.1007/BF02775440.  Google Scholar

[15]

E. Glasner and X. Ye, Local entropy theory,, Ergodic Theory and Dynam. Systems, 29 (2009), 321.  doi: 10.1017/S0143385708080309.  Google Scholar

[16]

E. Glasner, Classifying dynamical systems by their recurrence properties,, Topol. Methods Nonlinear Anal., 24 (2004), 21.   Google Scholar

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W. Huang, S. Shao and X. Ye, Mixing and proximal cells along a sequences,, Nonlinearity, 17 (2004), 1245.  doi: 10.1088/0951-7715/17/4/006.  Google Scholar

[18]

W. Huang and X. Ye, Dynamical systems disjoint from and minimal system,, Tran. Amer. Math. Soc., 357 (2005), 669.  doi: 10.1090/S0002-9947-04-03540-8.  Google Scholar

[19]

W. Huang and X. Ye, Topological complexity, return times and weak disjointness,, Ergod. Thero. Dyn. Syst., 24 (2004), 825.  doi: 10.1017/S0143385703000543.  Google Scholar

[20]

W. Huang and X. Ye, A local variational relation and applications,, Israel J. Math., 151 (2006), 237.  doi: 10.1007/BF02777364.  Google Scholar

[21]

A. Illanes and S. Nadler, Hyperspaces,, Fundamentals and Recent Advances, (1999).   Google Scholar

[22]

J. Li, Transitive points via Furstenberg family,, Topology Appl., 158 (2011), 2221.  doi: 10.1016/j.topol.2011.07.013.  Google Scholar

[23]

J. Li, P. Oprocha and G. Zhang, On recurrence over subsets and weak mixing,, preprint, (2013).   Google Scholar

[24]

J. Mycielski, Independent sets in topological algebras,, Fund. Math., 55 (1964), 139.   Google Scholar

[25]

P. Oprocha, Coherent lists and chaotic sets,, Discrete Continuous Dynam. Systems, 31 (2011), 797.  doi: 10.3934/dcds.2011.31.797.  Google Scholar

[26]

P. Oprocha and G. Zhang, On local aspects of topological weak mixing in dimension one and beyond,, Studia Math., 202 (2011), 261.  doi: 10.4064/sm202-3-4.  Google Scholar

[27]

P. Oprocha and G. Zhang, On sets with recurrence properties, their topological structure and entropy,, Top. App., 159 (2012), 1767.  doi: 10.1016/j.topol.2011.04.020.  Google Scholar

[28]

P. Oprocha and G. Zhang, On weak product recurrence and synchroniztion of return times,, Adv. Math., 244 (2013), 395.  doi: 10.1016/j.aim.2013.05.006.  Google Scholar

[29]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

[30]

J. Xiong and Z. Yang, Chaos caused by a toplogical mixing map,, in Dynamical Systems and Related Topics (Nagoya, (1990), 550.   Google Scholar

show all references

References:
[1]

E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions,, The University Series in Mathematics, (1997).  doi: 10.1007/978-1-4757-2668-8.  Google Scholar

[2]

E. Akin, Lectures on Cantor and Mycielski sets for dynamical systems,, in Chapel Hill Ergodic Theory Workshops, (2004), 21.  doi: 10.1090/conm/356/06496.  Google Scholar

[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.  doi: 10.1017/S0143385709000753.  Google Scholar

[4]

F. Blanchard, Fully positive topological entropy and topological mixing,, Symbolic Dynamics and its Applications (New Haven, (1991), 95.  doi: 10.1090/conm/135/1185082.  Google Scholar

[5]

F. Blanchard, A disjointness theorem involving topological entropy,, Bull. Soc. Math. France, 121 (1993), 465.   Google Scholar

[6]

F. Blanchard, B. Host, A. Maass, S. Martinez and D. Rudolph, Entropy pairs for a measure,, Ergod. Theory Dynam. Syst., 15 (1995), 621.  doi: 10.1017/S0143385700008579.  Google Scholar

[7]

F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity,, Discrete Contin. Dyn. Syst., 20 (2008), 275.   Google Scholar

[8]

D. Dou, X. Ye and G. Zhang, Entropy sequence and maximal entropy sets,, Nonlinearity, 19 (2006), 53.  doi: 10.1088/0951-7715/19/1/004.  Google Scholar

[9]

R. Ellis, Extending continuous functions on zero-dimensional spaces,, Math. Ann., 186 (1970), 114.  doi: 10.1007/BF01350686.  Google Scholar

[10]

R. Ellis, S. Glasner and L. Shapiro, Proximal-Isometric Flows,, Advances in Math., 17 (1975), 213.  doi: 10.1016/0001-8708(75)90093-6.  Google Scholar

[11]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, M. B. Porter Lectures, (1981).   Google Scholar

[12]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Collooquium Publications, (1955).   Google Scholar

[13]

E. Glasner, A simple characterization of the set of $\mu$-entropy pairs and applications,, Israel J. Math., 102 (1997), 13.  doi: 10.1007/BF02773793.  Google Scholar

[14]

E. Glasner, Topological weak mixing and quasi-Bohr systems,, Israel J. Math., 148 (2005), 277.  doi: 10.1007/BF02775440.  Google Scholar

[15]

E. Glasner and X. Ye, Local entropy theory,, Ergodic Theory and Dynam. Systems, 29 (2009), 321.  doi: 10.1017/S0143385708080309.  Google Scholar

[16]

E. Glasner, Classifying dynamical systems by their recurrence properties,, Topol. Methods Nonlinear Anal., 24 (2004), 21.   Google Scholar

[17]

W. Huang, S. Shao and X. Ye, Mixing and proximal cells along a sequences,, Nonlinearity, 17 (2004), 1245.  doi: 10.1088/0951-7715/17/4/006.  Google Scholar

[18]

W. Huang and X. Ye, Dynamical systems disjoint from and minimal system,, Tran. Amer. Math. Soc., 357 (2005), 669.  doi: 10.1090/S0002-9947-04-03540-8.  Google Scholar

[19]

W. Huang and X. Ye, Topological complexity, return times and weak disjointness,, Ergod. Thero. Dyn. Syst., 24 (2004), 825.  doi: 10.1017/S0143385703000543.  Google Scholar

[20]

W. Huang and X. Ye, A local variational relation and applications,, Israel J. Math., 151 (2006), 237.  doi: 10.1007/BF02777364.  Google Scholar

[21]

A. Illanes and S. Nadler, Hyperspaces,, Fundamentals and Recent Advances, (1999).   Google Scholar

[22]

J. Li, Transitive points via Furstenberg family,, Topology Appl., 158 (2011), 2221.  doi: 10.1016/j.topol.2011.07.013.  Google Scholar

[23]

J. Li, P. Oprocha and G. Zhang, On recurrence over subsets and weak mixing,, preprint, (2013).   Google Scholar

[24]

J. Mycielski, Independent sets in topological algebras,, Fund. Math., 55 (1964), 139.   Google Scholar

[25]

P. Oprocha, Coherent lists and chaotic sets,, Discrete Continuous Dynam. Systems, 31 (2011), 797.  doi: 10.3934/dcds.2011.31.797.  Google Scholar

[26]

P. Oprocha and G. Zhang, On local aspects of topological weak mixing in dimension one and beyond,, Studia Math., 202 (2011), 261.  doi: 10.4064/sm202-3-4.  Google Scholar

[27]

P. Oprocha and G. Zhang, On sets with recurrence properties, their topological structure and entropy,, Top. App., 159 (2012), 1767.  doi: 10.1016/j.topol.2011.04.020.  Google Scholar

[28]

P. Oprocha and G. Zhang, On weak product recurrence and synchroniztion of return times,, Adv. Math., 244 (2013), 395.  doi: 10.1016/j.aim.2013.05.006.  Google Scholar

[29]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

[30]

J. Xiong and Z. Yang, Chaos caused by a toplogical mixing map,, in Dynamical Systems and Related Topics (Nagoya, (1990), 550.   Google Scholar

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