# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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.  Google Scholar [2] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.  Google Scholar [3] J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153 North-Holland, Amsterdam, 1988.  Google Scholar [4] L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, Springer, 2012. doi: 10.1007/978-3-642-28090-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] 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.  Google Scholar [12] W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.  doi: 10.24033/bsmf.2534.  Google Scholar [13] W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.  Google Scholar [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.  Google Scholar [15] J. Li, S. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.  Google Scholar [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.  Google Scholar [17] S. Li, $ω$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243-249.  doi: 10.1090/S0002-9947-1993-1108612-8.  Google Scholar [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. Google Scholar [19] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79 Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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##### 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.  Google Scholar [2] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.  Google Scholar [3] J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153 North-Holland, Amsterdam, 1988.  Google Scholar [4] L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, Springer, 2012. doi: 10.1007/978-3-642-28090-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] 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.  Google Scholar [12] W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282.  doi: 10.24033/bsmf.2534.  Google Scholar [13] W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.  Google Scholar [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.  Google Scholar [15] J. Li, S. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.  Google Scholar [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.  Google Scholar [17] S. Li, $ω$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243-249.  doi: 10.1090/S0002-9947-1993-1108612-8.  Google Scholar [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. Google Scholar [19] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79 Springer-Verlag, New York-Berlin, 1982.  Google Scholar
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