Dynamics on sensitive and equicontinuous functions

Jie Li, Tao Yu, Tiaoying Zeng

DOI: http://dx.doi.org/10.12775/TMNA.2017.054

Abstract


The notions of sensitive and equicontinuous functions under semigroup action are introduced and intensively studied. We show that a transitive system is sensitive if and only if it has a sensitive pair if and only if it has a sensitive function. While there exists a minimal non-weakly mixing system such that every non-constant continuous function is sensitive, and a topological dynamical system is weakly mixing if and only if it is sensitive consistently with respect to (at least) any two non-constant continuous functions. We also get a dichotomy result for minimal systems -- every continuous function is either sensitive or equicontinuous.

Keywords


Sensitivity; sensitive pairs; sensitive functions; equicontinuous functions; weak mixing

Full Text:

PREVIEW FULL TEXT

References


M. Achigar, A. Artigue and I. Monteverde, Observing expansive maps, arXiv:1611.08488v1, 2016.

E. Akin, J. Auslander and E. Glasner, The topological dynamics of Ellis actions, Mem. Amer. Math. Soc. 195 (2008), no. 913, vi+152 pp.

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

J. Auslander and J. Yorke, Interval maps, factors of maps and chaos, Tôhoku Math. J. (2) 32 (1980), no. 2, 177–188.

F. Garcı́a-Ramos and B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, arXiv:1509.05246v3, 2015.

E. Glasner and M. Megrelishvili, Linear representations of hereditarily non-sensitive dynamical systems, Colloq. Math. 104 (2006), no. 2, 223–283.

E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems, J. Amer. Math. Soc. 8 (1995), 665–686.

D. Kerr and H. Li, Dynamical entropy in banach spaces, Invent. Math. 162 (2005), no. 3, 649–686.

D. Kerr and H. Li, Independence in topological and C ∗ -dynamics, Math. Ann. 338 (2007), 869–926.

E. Kontorovich and M. Megrelishvili, A note on sensitivity of semigroup actions, Semigroup Forum 76 (2008), no. 1, 133–141.

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 1, 83–114.

K. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc. 24 (1970), 278–280.

D. Ruelle, Dynamical systems with turbulent behavior, Mathematical Problems in Theoretical Physics, Proc. Internat. Conf., Univ. Rome, Rome, 1977, Lecture Notes in Phys., vol. 80, Springer, Berlin, New York, 1978, pp. 341–360.

S. Shao, X. Ye and R. Zhang, Sensitivity and regionally proximal relation in minimal systems, Sci. China Math. 51 (2008), 987–994.

H. Wang, Z. Chen and H. Fu, M-systems and scattering systems of semigroup actions, Semigroup Forum 91 (2015), no. 3, 699–717.

J. Xiong, Chaos in a topologically transitive system, Sci. China Ser. A 48 (2005), 929–939.

X. Ye and R. Zhang, On sensitive sets in topological dynamics, Nonlinearity 21 (2008), 1601–1620.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism