Equicontinuity and Sensitivity in Mean Forms

In this paper two mean forms of equicontinuity and sensitivity are mainly studied. For the equicontinuity side, it is shown that any topological dynamical system can be embedded into some almost equicontinuous in the mean system. For the sensitivity side, it turns out that the notions of sensitivity in the mean and mean sensitivity are identical in the measure-theoretical setting, though they are different in the topological setting. Moreover, levels of mean multi-variant versions of sensitivity in topological or measure-theoretical sense are completely classified.

[1]  Pa Hulse.  Sequence Entropy and Subsequence Generators , 1982 .

[2]  Measurable sensitivity via Furstenberg families , 2017 .

[3]  Jiahao Qiu,et al.  A Note on Mean Equicontinuity , 2018, Journal of Dynamics and Differential Equations.

[4]  Wolfgang Krieger.  On unique ergodicity , 1972 .

[5]  S. Glasner,et al.  Rigidity in topological dynamics , 1989, Ergodic Theory and Dynamical Systems.

[6]  Jie-lin Li.  How chaotic is an almost mean equicontinuous system , 2018 .

[7]  J. Auslander,et al.  Minimal flows and their extensions , 1988 .

[8]  Jian Li,et al.  Stable sets and mean Li–Yorke chaos in positive entropy systems , 2012, 1211.6836.

[9]  Wen Huang,et al.  Measure-theoretical sensitivity and equicontinuity , 2011 .

[10]  Jian Li.  Measure-theoretic sensitivity via finite partitions , 2016, 1606.03649.

[11]  X. Ye,et al.  Recent development of chaos theory in topological dynamics , 2015, 1503.06425.

[12]  X. Ye,et al.  Mean equicontinuity and mean sensitivity , 2013, Ergodic Theory and Dynamical Systems.

[13]  Felipe García-Ramos.  Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy , 2016, Ergodic Theory and Dynamical Systems.

[14]  Xiangdong Ye,et al.  A local variational relation and applications , 2006 .

[15]  Harry Furstenberg,et al.  Recurrence in Ergodic Theory and Combinatorial Number Theory , 2014 .

[16]  Xiangdong Ye,et al.  Mean equicontinuity, almost automorphy and regularity , 2019, Israel Journal of Mathematics.

[17]  WEN HUANG,et al.  Bounded complexity, mean equicontinuity and discrete spectrum , 2018, Ergodic Theory and Dynamical Systems.

[18]  Wen Huang,et al.  Combinatorial lemmas and applications to dynamics , 2009 .

[19]  James A. Yorke,et al.  INTERVAL MAPS, FACTORS OF MAPS, AND CHAOS , 1980 .

[20]  Xiong Jincheng.  Chaos in a topologically transitive system , 2005 .

[21]  J. Neumann,et al.  Operator Methods in Classical Mechanics, II , 1942 .

[22]  de Jan Vries,et al.  Elements of Topological Dynamics , 1993 .

[23]  Felipe Garc'ia-Ramos,et al.  When is a dynamical system mean sensitive? , 2017, Ergodic Theory and Dynamical Systems.

[24]  Tao Yu,et al.  Mean equicontinuity, complexity and applications , 2021 .

[25]  Ethan Akin,et al.  When is a Transitive Map Chaotic , 1996 .

[26]  X. Ye,et al.  On sensitive sets in topological dynamics , 2008 .