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March  2015, 35(3): 1059-1073. doi: 10.3934/dcds.2015.35.1059

Recurrence properties and disjointness on the induced spaces

Jie Li 1, , Kesong Yan 1, and  Xiangdong Ye 1,

1. 

Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China, China, China

Received  February 2014 Revised  August 2014 Published  October 2014

A topological dynamical system induces two natural systems, one is on the hyperspace and the other one is on the probability measures space. The connection among some dynamical properties on the original space and on the induced spaces are investigated. Particularly, a minimal weakly mixing system which induces a $P$-system on the probability measures space is constructed and some disjointness result is obtained.
Keywords: weakly mixing, minimal, Induced spaces, disjointness..
Mathematics Subject Classification: Primary: 54B20; Secondary: 54H2.
Citation: Jie Li, Kesong Yan, Xiangdong Ye. Recurrence properties and disjointness on the induced spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1059-1073. doi: 10.3934/dcds.2015.35.1059
References:
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J. Auslander, Minimal Flows and Their Extensions,, North-Holland Mathematics Studies, 153 (1988).   Google Scholar

[2]

J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Th. and Dynam. Sys., 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

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J. Banks, Chaos for induced hyperspace maps,, Chaos Solitons Fractals, 25 (2005), 681.  doi: 10.1016/j.chaos.2004.11.089.  Google Scholar

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W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures,, Monatsh. Math., 79 (1975), 81.  doi: 10.1007/BF01585664.  Google Scholar

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A. Blokh and A. Fieldsteel, Sets that force recurrence,, Proc. Amer. Math. Soc., 130 (2002), 3571.  doi: 10.1090/S0002-9939-02-06349-9.  Google Scholar

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M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces,, Lecture Notes in Mathematics, 527 (1976).  doi: 10.1007/BFb0082364.  Google Scholar

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P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness,, Israel J. of Math., 188 (2012), 463.  doi: 10.1007/s11856-011-0128-z.  Google Scholar

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H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory , 1 (1967), 1.  doi: 10.1007/BF01692494.  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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E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems,, J. Amer. Math. Soc., 8 (1995), 665.  doi: 10.1090/S0894-0347-1995-1270579-5.  Google Scholar

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J. Guirao, K. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hyperspaces,, Nonlinear Anal., 71 (2009), 1.  doi: 10.1016/j.na.2008.10.055.  Google Scholar

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W. Huang, H. Li and X. Ye, Family independence for topological and measurable dynamics,, Trans. Amer. Math. Soc., 364 (2012), 5209.  doi: 10.1090/S0002-9947-2012-05493-6.  Google Scholar

[13]

W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems,, Bull. Soc. Math. France, 135 (2007), 259.   Google Scholar

[14]

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

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D. Kerr and H. Li, Dynamical entropy in Banach spaces,, Invent. Math., 162 (2005), 649.  doi: 10.1007/s00222-005-0457-9.  Google Scholar

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D. Kerr and H. Li, Independence in topological and $C^*$-dynamics,, Math. Ann., 338 (2007), 869.  doi: 10.1007/s00208-007-0097-z.  Google Scholar

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M. Komuro, The pseudo orbit tracing properties on the space of probability measures,, Tokyo J. Math., 7 (1984), 461.  doi: 10.3836/tjm/1270151738.  Google Scholar

[18]

E. Lehrer, Topological mixing and uniquely ergodic systems,, Israel J. Math., 57 (1987), 239.  doi: 10.1007/BF02772176.  Google Scholar

[19]

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

[20]

J. Li, Equivalent conditions of Devaney chaos on the hyperspace,, J. Univ. Sci. Technol. China, 44 (2014), 93.   Google Scholar

[21]

Z. Lian, S. Shao and X. Ye, Weakly mixing, proximal topological models for ergodic systems and applications,, preprint, ().   Google Scholar

[22]

Sam B. Nadler, Jr., Continuum Theory: An Introduction,, Pure and Applied Mathematics, 158 (1992).   Google Scholar

[23]

P. Oprocha, Weak mixing and product recurrence,, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233.  doi: 10.5802/aif.2553.  Google Scholar

[24]

K. R. Parthasarathy, Probability Measures on Metric Spaces,, Probability and Mathematical Statistics, 3 (1967).   Google Scholar

[25]

K. Petersen, Disjointness and weak mixing of minimal sets,, Proc. Amer. Math. Soc., 24 (1970), 278.  doi: 10.1090/S0002-9939-1970-0250283-7.  Google Scholar

[26]

H. Román-Flores, A note on transitivity in set-valued discrete systems,, Chaos Solitons Fractals, 17 (2003), 99.  doi: 10.1016/S0960-0779(02)00406-X.  Google Scholar

[27]

S. Shao, Dynamical Systems and Families,, PhD thesis, (2003).   Google Scholar

[28]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, 79 (1982).   Google Scholar

show all references

References:
[1]

J. Auslander, Minimal Flows and Their Extensions,, North-Holland Mathematics Studies, 153 (1988).   Google Scholar

[2]

J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Th. and Dynam. Sys., 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

[3]

J. Banks, Chaos for induced hyperspace maps,, Chaos Solitons Fractals, 25 (2005), 681.  doi: 10.1016/j.chaos.2004.11.089.  Google Scholar

[4]

W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures,, Monatsh. Math., 79 (1975), 81.  doi: 10.1007/BF01585664.  Google Scholar

[5]

A. Blokh and A. Fieldsteel, Sets that force recurrence,, Proc. Amer. Math. Soc., 130 (2002), 3571.  doi: 10.1090/S0002-9939-02-06349-9.  Google Scholar

[6]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces,, Lecture Notes in Mathematics, 527 (1976).  doi: 10.1007/BFb0082364.  Google Scholar

[7]

P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness,, Israel J. of Math., 188 (2012), 463.  doi: 10.1007/s11856-011-0128-z.  Google Scholar

[8]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory , 1 (1967), 1.  doi: 10.1007/BF01692494.  Google Scholar

[9]

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

[10]

E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems,, J. Amer. Math. Soc., 8 (1995), 665.  doi: 10.1090/S0894-0347-1995-1270579-5.  Google Scholar

[11]

J. Guirao, K. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hyperspaces,, Nonlinear Anal., 71 (2009), 1.  doi: 10.1016/j.na.2008.10.055.  Google Scholar

[12]

W. Huang, H. Li and X. Ye, Family independence for topological and measurable dynamics,, Trans. Amer. Math. Soc., 364 (2012), 5209.  doi: 10.1090/S0002-9947-2012-05493-6.  Google Scholar

[13]

W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems,, Bull. Soc. Math. France, 135 (2007), 259.   Google Scholar

[14]

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

[15]

D. Kerr and H. Li, Dynamical entropy in Banach spaces,, Invent. Math., 162 (2005), 649.  doi: 10.1007/s00222-005-0457-9.  Google Scholar

[16]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics,, Math. Ann., 338 (2007), 869.  doi: 10.1007/s00208-007-0097-z.  Google Scholar

[17]

M. Komuro, The pseudo orbit tracing properties on the space of probability measures,, Tokyo J. Math., 7 (1984), 461.  doi: 10.3836/tjm/1270151738.  Google Scholar

[18]

E. Lehrer, Topological mixing and uniquely ergodic systems,, Israel J. Math., 57 (1987), 239.  doi: 10.1007/BF02772176.  Google Scholar

[19]

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

[20]

J. Li, Equivalent conditions of Devaney chaos on the hyperspace,, J. Univ. Sci. Technol. China, 44 (2014), 93.   Google Scholar

[21]

Z. Lian, S. Shao and X. Ye, Weakly mixing, proximal topological models for ergodic systems and applications,, preprint, ().   Google Scholar

[22]

Sam B. Nadler, Jr., Continuum Theory: An Introduction,, Pure and Applied Mathematics, 158 (1992).   Google Scholar

[23]

P. Oprocha, Weak mixing and product recurrence,, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233.  doi: 10.5802/aif.2553.  Google Scholar

[24]

K. R. Parthasarathy, Probability Measures on Metric Spaces,, Probability and Mathematical Statistics, 3 (1967).   Google Scholar

[25]

K. Petersen, Disjointness and weak mixing of minimal sets,, Proc. Amer. Math. Soc., 24 (1970), 278.  doi: 10.1090/S0002-9939-1970-0250283-7.  Google Scholar

[26]

H. Román-Flores, A note on transitivity in set-valued discrete systems,, Chaos Solitons Fractals, 17 (2003), 99.  doi: 10.1016/S0960-0779(02)00406-X.  Google Scholar

[27]

S. Shao, Dynamical Systems and Families,, PhD thesis, (2003).   Google Scholar

[28]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, 79 (1982).   Google Scholar

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