Recurrence properties and disjointness on the induced spaces

Previous Article
On the integral systems with negative exponents
DCDS Home
This Issue
Next Article
Center conditions for a class of planar rigid polynomial differential systems

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

Abstract
Related Papers

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:

[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

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

[1]	Piotr Oprocha. Double minimality, entropy and disjointness with all minimal systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 263-275. doi: 10.3934/dcds.2019011
[2]	Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
[3]	François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[4]	Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
[5]	Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
[6]	Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629
[7]	Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067
[8]	Ming Wang. Global attractor for weakly damped gKdV equations in higher sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3799-3825. doi: 10.3934/dcds.2015.35.3799
[9]	Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure & Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301
[10]	Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315
[11]	Philipp Kunde. Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions. Journal of Modern Dynamics, 2016, 10: 439-481. doi: 10.3934/jmd.2016.10.439
[12]	Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53
[13]	Karim Boulabiar, Gerard Buskes and Gleb Sirotkin. A strongly diagonal power of algebraic order bounded disjointness preserving operators. Electronic Research Announcements, 2003, 9: 94-98.
[14]	Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[15]	Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010
[16]	Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082
[17]	A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195
[18]	Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259
[19]	Matthew Nicol. Induced maps of hyperbolic Bernoulli systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 147-154. doi: 10.3934/dcds.2001.7.147
[20]	Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences