Some remarks on chaos in topological dynamics

Huoyung Wang, Heman Fu
DOI: https://doi.org/10.4995/agt.2011.1645

Abstract

Bau-Sen Du introduced a notion of chaos which is stronger than Li-Yorke sensitivity. A TDS (X, f) is called chaotic if there is a positive e such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not e-asymptotic. In this article, we show that a TDS (T, f) is transitive but not mixing if and only if (T, f) is Li-Yorke sensitive but not chaotic, where T is a tree. Moreover, we compare such chaos with other notions of chaos.

Keywords

Sensitivity; Chaos; Tree maps

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References

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003), 1421–1433. http://dx.doi.org/10.1088/0951-7715/16/4/313

L. Alseda, S. Kolyada, J. Llibre and L. Snoha, Entropy and Periodic points for tree maps, Trans. Amer. Math. Soc. 351 (1997), 1551–1573. http://dx.doi.org/10.1090/S0002-9947-99-02077-2

R. Devaney, Chaotic Dynamical Systems, Addison-Wesley, Redwood City, 1980.

B. Du, On the nature of chaos, arXiv: math.DS/0602585 v1 26 Feb 2006.

T. Li and J. Yorke, Period 3 implies chaos, Amer. Math. Monthly 82 (1975), 985–992. http://dx.doi.org/10.2307/2318254

J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147.

L. Wang, Z. Chen and G. Liao, The complexity of a minimal sub-shift on symbolic spaces, J. Math. Anal. Appl. 37 (2006), 136–145. http://dx.doi.org/10.1016/j.jmaa.2005.12.069

X. Ye, The center and the depth of the center of a tree map, Bull. Austral. Math. Soc. 48 (1993), 347–350. http://dx.doi.org/10.1017/S0004972700015768

X. Ye, W. Huang and S. Shao, An Introduction to Topolgical Dynamics, Science Press, Bejing, 2008. [Chinese]

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