Skip to main content Accessibility help
×
×
Home

Mean equicontinuity and mean sensitivity

  • JIAN LI (a1), SIMING TU (a2) and XIANGDONG YE (a2)

Abstract

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

Copyright

References

Hide All
[1]Akin, E., Auslander, J. and Berg, K.. When is a transitive map chaotic? Convergence in Ergodic Theory and Probability (Columbus, OH, 1993) (Ohio State University Mathematics Research Institute Publication, 5). de Gruyter, Berlin, 1996, pp. 2540.
[2]Auslander, J.. Mean-L-stable systems. Illinois J. Math. 3 (1959), 566579.
[3]Auslander, J.. Minimal Flows and Their Extensions. North-Holland Publishing Co., Amsterdam, 1988.
[4]Auslander, J. and Yorke, J. A.. Interval maps, factors of maps, and chaos. Tohoku Math. J. 32(2) (1980), 177188.
[5]Blanchard, F.. A disjointness theorem involving topological entropy. Bull. Soc. Math. France 121 (1993), 465478.
[6]Downarowicz, T.. Positive topological entropy implies chaos DC2. Proc. Amer. Math. Soc. 142 (2014), 137149.
[7]Ellis, R. and Gottschalk, W. H.. Homomorphisms of transformation groups. Trans. Amer. Math. Soc. 94 (1960), 258271.
[8]Fomin, S.. On dynamical systems with a purely point spectrum. Dokl. Akad. Nauk SSSR 77 (1951), 2932 ; In Russian.
[9]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.
[10]García-Ramos, F.. Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy. Preprint, 2010, arXiv:1402.7327V2 [MATH.DS].
[11]Gillis, J.. Notes on a property of measurable sets. J. Lond. Math. Soc. 11 (1936), 139141.
[12]Glasner, E.. The structure of tame minimal dynamical systems. Ergod. Th. & Dynam. Sys. 27 (2007), 18191837.
[13]Glasner, S. and Maon, D.. Rigidity in topological dynamics. Ergod. Th. & Dynam. Sys. 9 (1989), 309320.
[14]Glasner, E., Megrelishvili, M. and Uspenskij, V.. On metrizable enveloping semigroups. Israel J. Math. 164 (2008), 317332.
[15]Glasner, E. and Weiss, B.. Sensitive dependence on initial conditions. Nonlinearity 6(6) (1993), 10671075.
[16]Hahn, F. and Katznelson, Y.. On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc. 126 (1967), 335360.
[17]Huang, W., Li, S., Shao, S. and Ye, X.. Null systems and sequence entropy pairs. Ergod Th. & Dynam. Sys. 23 (2003), 15051523.
[18]Huang, W., Li, J. and Ye, X.. Stable sets and mean Li–York chaos in positive entropy systems. J. Funct. Anal. 266 (2014), 33773394.
[19]Huang, W., Lu, P. and Ye, X.. Measure-theoretical sensitivity and equicontinuity. Israel J. Math. 183 (2011), 233283.
[20]Huang, W. and Ye, X.. Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl. 117 (2002), 259272.
[21]Huang, W. and Ye, X.. Minimal sets in almost equicontinuous systems. Proc. Steklov Inst. Math. 244 (2004), 280287.
[22]Huang, W. and Ye, X.. A local variational relation and applications. Israel J. Math. 151 (2006), 237280.
[23]Kerr, D. and Li, H.. Independence in topological and C -dynamics. Math. Ann. 338 (2007), 869926.
[24]Li, J. and Tu, S.. On proximality with Banach density one. J. Math. Anal. Appl. 416 (2014), 3651.
[25]Ornstein, D. and Weiss, B.. Mean distality and tightness. Proc. Steklov Inst. Math. 244(1) (2004), 295302.
[26]Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116136.
[27]Scarpellini, B.. Stability properties of flows with pure point spectrum. J. Lond. Math. Soc. (2) 26(3) (1982), 451464.
[28]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York-Berlin, 1982.
[29]Ye, X. and Zhang, R.. On sensitivity sets in topological dynamics. Nonlinearity 21 (2008), 16011620.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed