Stronger Version Sensitivity, Almost Finite to One Extension and Maximal Pattern Entropy

Abstract

In this paper, we introduce thick r-sensitivity, multi-r-sensitivity and block thick r-sensitivity for $r\ge 2$. We first give a characterization of a minimal system which is block thickly r-sensitive. Then we obtain a sufficient condition of a minimal system which is thickly r-sensitive. The maximal pattern entropy of a multi-r-sensitive topological dynamical system is also discussed.

Keywords

Sensitivity / Minimal system / Almost finite to one extension / Maximal type entropy

References

[1] Abraham C, Biau G, Cadre B. On Lyapunov exponent and sensitivity. J. Math. Anal. Appl.. 2004, 290 2 395-404

[2] Akin E. Recurrence in Topological Dynamics, The University Series in Mathematics. 1997 New York: Plenum Press

[3] Akin E, Kolyada S. Li-Yorke sensitivity. Nonlinearity. 2003, 16 4 1421-1433

[4] Auslander J. Minimal Flows and Their Extensions, North-Holland Mathematics Studies, Notas de Matemática [Mathematical Notes], 122. 1988 Amsterdam: North-Holland Publishing Co.

[5] Auslander J, Yorke JA. Interval maps, factors of maps, and chaos. Tôhoku Math. J. (2). 1980, 32 2 177-188

[6] Blanchard F, Host B, Maass A. Topological complexity. Ergod. Theory Dyn. Syst.. 2000, 20 3 641-662

[7] Ellis R, Gottschalk WH. Homomorphisms of transformation groups. Trans. Am. Math. Soc.. 1960, 94 258-271

[8] Furstenberg H. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory. 1967, 1 1-49

[9] Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. 1981 Princeton: Princeton University Press

[10] Glasner E, Weiss B. Sensitive dependence on initial conditions. Nonlinearity. 1993, 6 6 1067-1075

[11] Guckenheimer J. Sensitive dependence to initial conditions for one-dimensional maps. Commun. Math. Phys.. 1979, 70 2 133-160

[12] Huang W, Khilko D, Kolyada S, Zhang GH. Dynamical compactness and sensitivity. J. Differ. Equ.. 2016, 260 9 6800-6827

[13] Huang, W., Kolyada, S., Zhang, G.H.: Analogues of Auslander-Yorke theorems for multi-sensitivity. Ergod. Th. and Dynam. Sys. (to appear)

[14] Huang W, Lu P, Ye X. Measure-theoretical sensitivity and equicontinuity. Isr. J. Math.. 2011, 183 233-283

[15] Huang W, Ye X. Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topol. Appl.. 2002, 117 3 259-272

[16] Huang W, Ye X. An explicit scattering, non-weakly mixing example and weak disjointness. Nonlinearity. 2002, 15 3 849-862

[17] Huang W, Ye X. Combinatorial lemmas and applications to dynamics. Adv. Math.. 2009, 220 6 1689-1716

[18] Liu, H., Liao, L., Wang, L.: Thickly syndetical sensitivity of topological dynamical system. Discrete Dyn. Nat. Soc. 4 (2014)

[19] Maass A, Shao S. Structure of bounded topological-sequence-entropy minimal systems. J. Lond. Math. Soc. (2). 2007, 76 3 702-718

[20] Shao S, Ye X, Zhang R. Sensitivity and regionally proximal relation in minimal systems. Sci. China Ser. A. 2008, 51 6 987-994

[21] Subrahmonian Moothathu TK. Stronger forms of sensitivity for dynamical systems. Nonlinearity. 2007, 20 9 2115-2126

[22] Ruelle D. Dynamical Systems with Turbulent Behavior, Mathematical Problems in Theoretical Physics (Proceedings of International Conference, University of Rome, 1977), Lecture Notes in Physics. 1978 Berlin: Springer

[23] Veech W. The equicontinuous structure relation for minimal Abelian transformation groups. Am. J. Math.. 1968, 90 723-732

[24] Xiong J. Chaos in topological transitive systems. Sci. China Ser. A Math.. 2005, 48 929-939

[25] Ye, X., Yu, T.: Sensitivity, proximal extension and higher order almost automorphy. Trans. Am. Math. Soc., to appear