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

Recurrence properties and disjointness on the induced spaces

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.
Citation: Jie Li, Kesong Yan, Xiangdong Ye. Recurrence properties and disjointness on the induced spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1059-1073. doi: 10.3934/dcds.2015.35.1059
References:
[1]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland, Amsterdam, 1988.

[2]

J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Th. and Dynam. Sys., 17 (1997), 505-529. doi: 10.1017/S0143385797069885.

[3]

J. Banks, Chaos for induced hyperspace maps, Chaos Solitons Fractals, 25 (2005), 681-685. doi: 10.1016/j.chaos.2004.11.089.

[4]

W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math., 79 (1975), 81-92. doi: 10.1007/BF01585664.

[5]

A. Blokh and A. Fieldsteel, Sets that force recurrence, Proc. Amer. Math. Soc., 130 (2002), 3571-3578. doi: 10.1090/S0002-9939-02-06349-9.

[6]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/BFb0082364.

[7]

P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. of Math., 188 (2012), 463-507. doi: 10.1007/s11856-011-0128-z.

[8]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory , 1 (1967), 1-49. doi: 10.1007/BF01692494.

[9]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.

[10]

E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686. doi: 10.1090/S0894-0347-1995-1270579-5.

[11]

J. Guirao, K. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hyperspaces, Nonlinear Anal., 71 (2009), 1-8. doi: 10.1016/j.na.2008.10.055.

[12]

W. Huang, H. Li and X. Ye, Family independence for topological and measurable dynamics, Trans. Amer. Math. Soc., 364 (2012), 5209-5242. doi: 10.1090/S0002-9947-2012-05493-6.

[13]

W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.

[14]

W. Huang and X. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694. doi: 10.1090/S0002-9947-04-03540-8.

[15]

D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686. doi: 10.1007/s00222-005-0457-9.

[16]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926. doi: 10.1007/s00208-007-0097-z.

[17]

M. Komuro, The pseudo orbit tracing properties on the space of probability measures, Tokyo J. Math., 7 (1984), 461-468. doi: 10.3836/tjm/1270151738.

[18]

E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239-255. doi: 10.1007/BF02772176.

[19]

J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231. doi: 10.1016/j.topol.2011.07.013.

[20]

J. Li, Equivalent conditions of Devaney chaos on the hyperspace, J. Univ. Sci. Technol. China, 44 (2014), 93-95.

[21]

Z. Lian, S. Shao and X. Ye, Weakly mixing, proximal topological models for ergodic systems and applications, preprint, arXiv:1407.1978v1.

[22]

Sam B. Nadler, Jr., Continuum Theory: An Introduction, Pure and Applied Mathematics, 158, Marcel Dekker Inc., New York, 1992.

[23]

P. Oprocha, Weak mixing and product recurrence, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233-1257. doi: 10.5802/aif.2553.

[24]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, 3, Academic Press, Inc. New York-London, 1967.

[25]

K. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280. doi: 10.1090/S0002-9939-1970-0250283-7.

[26]

H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals, 17 (2003), 99-104. doi: 10.1016/S0960-0779(02)00406-X.

[27]

S. Shao, Dynamical Systems and Families, PhD thesis, University of Science and Technology of China, 2003.

[28]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

show all references

References:
[1]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland, Amsterdam, 1988.

[2]

J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Th. and Dynam. Sys., 17 (1997), 505-529. doi: 10.1017/S0143385797069885.

[3]

J. Banks, Chaos for induced hyperspace maps, Chaos Solitons Fractals, 25 (2005), 681-685. doi: 10.1016/j.chaos.2004.11.089.

[4]

W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math., 79 (1975), 81-92. doi: 10.1007/BF01585664.

[5]

A. Blokh and A. Fieldsteel, Sets that force recurrence, Proc. Amer. Math. Soc., 130 (2002), 3571-3578. doi: 10.1090/S0002-9939-02-06349-9.

[6]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/BFb0082364.

[7]

P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. of Math., 188 (2012), 463-507. doi: 10.1007/s11856-011-0128-z.

[8]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory , 1 (1967), 1-49. doi: 10.1007/BF01692494.

[9]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.

[10]

E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686. doi: 10.1090/S0894-0347-1995-1270579-5.

[11]

J. Guirao, K. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hyperspaces, Nonlinear Anal., 71 (2009), 1-8. doi: 10.1016/j.na.2008.10.055.

[12]

W. Huang, H. Li and X. Ye, Family independence for topological and measurable dynamics, Trans. Amer. Math. Soc., 364 (2012), 5209-5242. doi: 10.1090/S0002-9947-2012-05493-6.

[13]

W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.

[14]

W. Huang and X. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694. doi: 10.1090/S0002-9947-04-03540-8.

[15]

D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686. doi: 10.1007/s00222-005-0457-9.

[16]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926. doi: 10.1007/s00208-007-0097-z.

[17]

M. Komuro, The pseudo orbit tracing properties on the space of probability measures, Tokyo J. Math., 7 (1984), 461-468. doi: 10.3836/tjm/1270151738.

[18]

E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239-255. doi: 10.1007/BF02772176.

[19]

J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231. doi: 10.1016/j.topol.2011.07.013.

[20]

J. Li, Equivalent conditions of Devaney chaos on the hyperspace, J. Univ. Sci. Technol. China, 44 (2014), 93-95.

[21]

Z. Lian, S. Shao and X. Ye, Weakly mixing, proximal topological models for ergodic systems and applications, preprint, arXiv:1407.1978v1.

[22]

Sam B. Nadler, Jr., Continuum Theory: An Introduction, Pure and Applied Mathematics, 158, Marcel Dekker Inc., New York, 1992.

[23]

P. Oprocha, Weak mixing and product recurrence, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233-1257. doi: 10.5802/aif.2553.

[24]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, 3, Academic Press, Inc. New York-London, 1967.

[25]

K. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280. doi: 10.1090/S0002-9939-1970-0250283-7.

[26]

H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals, 17 (2003), 99-104. doi: 10.1016/S0960-0779(02)00406-X.

[27]

S. Shao, Dynamical Systems and Families, PhD thesis, University of Science and Technology of China, 2003.

[28]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

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