September  2018, 38(9): 4727-4744. doi: 10.3934/dcds.2018208

How chaotic is an almost mean equicontinuous system?

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

2. 

Einstein Institute of Mathematics, Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel

Received  January 2018 Revised  March 2018 Published  June 2018

Fund Project: Research of Jie Li was supported by China Postdoctoral Science Foundation (Grant no. 2017M611026), NNSF of China (Grant no. 11701231), NSF of Jiangsu Province (Grant no. BK20170225) and Science Foundation of Jiangsu Normal University (Grant no. 17XLR011).

The question how chaotic is an almost mean equicontinuous system is addressed. It is shown that every topological dynamical system can be embedded into an almost mean equicontinuous system with the same entropy which is an almost one-to-one extension of some mean equicontinuous system. Besides, there is an almost mean equicontinous system that is topologically K and Devaney chaotic, and as this consequence we know that every ergodic measure of such a topologically K system does not have full support.

Citation: Jie Li. How chaotic is an almost mean equicontinuous system?. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4727-4744. doi: 10.3934/dcds.2018208
References:
[1]

E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, Convergence in Ergodic Theory and Probability, de Gruyter, Berlin, 5 (1996), 25–40.

[2]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.

[3]

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

[4]

L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, Springer, 2012. doi: 10.1007/978-3-642-28090-0.

[5]

F. Blanchard, Fully positive topological entropy and topological mixing, in Symbolic Dynamics and its Applications, Contemporary Mathematics, 135 (1992), 95–105. doi: 10.1090/conm/135/1185082.

[6]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[7]

R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity 2$^{\text{nd}}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989, Studies in Nonlinearity. Westview Press, Boulder, CO, 2003.

[8]

F. García-Ramos, Weak forms of topological and measure theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy, Ergodic Theory Dynam. Systems, 37 (2017), 1211-1237.  doi: 10.1017/etds.2015.83.

[9]

F. García-Ramos, J. Li and R. Zhang, When is a dynamical system mean sensitive?, Ergodic Theory Dynam. Systems, 2017, arXiv: 1708.01987. doi: 10.1017/etds.2017.101.

[10]

P. Halmos and J. Von Neumann, Operator methods in classical mechanics, Ⅱ, Ann. of Math. (2), 43 (1942), 332-350.  doi: 10.2307/1968872.

[11]

W. HuangH. 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.

[12]

W. HuangK. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.  doi: 10.24033/bsmf.2534.

[13]

W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.

[14]

J. Li and S. Tu, On proximality with Banach density one, J. Math. Anal. Appl., 416 (2014), 36-51.  doi: 10.1016/j.jmaa.2014.02.021.

[15]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.

[16]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114.  doi: 10.1007/s10114-015-4574-0.

[17]

S. Li, $ω$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243-249.  doi: 10.1090/S0002-9947-1993-1108612-8.

[18]

S. Tu, Some Notions of Topological Dynamics in the Mean Sense, Nilsystem and Generalised Polynomial, Ph. D thesis, University of Science and Technology of China, 2014.

[19]

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

show all references

References:
[1]

E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, Convergence in Ergodic Theory and Probability, de Gruyter, Berlin, 5 (1996), 25–40.

[2]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.

[3]

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

[4]

L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, Springer, 2012. doi: 10.1007/978-3-642-28090-0.

[5]

F. Blanchard, Fully positive topological entropy and topological mixing, in Symbolic Dynamics and its Applications, Contemporary Mathematics, 135 (1992), 95–105. doi: 10.1090/conm/135/1185082.

[6]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[7]

R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity 2$^{\text{nd}}$ edition, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989, Studies in Nonlinearity. Westview Press, Boulder, CO, 2003.

[8]

F. García-Ramos, Weak forms of topological and measure theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy, Ergodic Theory Dynam. Systems, 37 (2017), 1211-1237.  doi: 10.1017/etds.2015.83.

[9]

F. García-Ramos, J. Li and R. Zhang, When is a dynamical system mean sensitive?, Ergodic Theory Dynam. Systems, 2017, arXiv: 1708.01987. doi: 10.1017/etds.2017.101.

[10]

P. Halmos and J. Von Neumann, Operator methods in classical mechanics, Ⅱ, Ann. of Math. (2), 43 (1942), 332-350.  doi: 10.2307/1968872.

[11]

W. HuangH. 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.

[12]

W. HuangK. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.  doi: 10.24033/bsmf.2534.

[13]

W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.

[14]

J. Li and S. Tu, On proximality with Banach density one, J. Math. Anal. Appl., 416 (2014), 36-51.  doi: 10.1016/j.jmaa.2014.02.021.

[15]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.

[16]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114.  doi: 10.1007/s10114-015-4574-0.

[17]

S. Li, $ω$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243-249.  doi: 10.1090/S0002-9947-1993-1108612-8.

[18]

S. Tu, Some Notions of Topological Dynamics in the Mean Sense, Nilsystem and Generalised Polynomial, Ph. D thesis, University of Science and Technology of China, 2014.

[19]

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

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