Devaney’s chaos implies existence of 𝑠-scrambled sets

Mai, Jie-Hua

doi:10.1090/S0002-9939-04-07514-8

Devaney’s chaos implies existence of $s$-scrambled sets
HTML articles powered by AMS MathViewer

by Jie-Hua Mai PDF

Proc. Amer. Math. Soc. 132 (2004), 2761-2767 Request permission

Abstract:

Let $X$ be a complete metric space without isolated points, and let $f:X\to X$ be a continuous map. In this paper we prove that if $f$ is transitive and has a periodic point of period $p$, then $f$ has a scrambled set $S=\bigcup _{n=1}^{\infty }C_{n}$ consisting of transitive points such that each $C_{n}$ is a synchronously proximal Cantor set, and $\bigcup _{i=0}^{p-1}f^{i}(S)$ is dense in $X$. Furthermore, if $f$ is sensitive (for example, if $f$ is chaotic in the sense of Devaney), with $2s$ being a sensitivity constant, then this $S$ is an $s$-scrambled set.

References

Ethan Akin, Joseph Auslander, and Kenneth Berg, When is a transitive map chaotic?, Convergence in ergodic theory and probability (Columbus, OH, 1993) Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 25–40. MR 1412595
J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (1992), no. 4, 332–334. MR 1157223, DOI 10.2307/2324899
Nilson C. Bernardes Jr., On the set of points with a dense orbit, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3421–3423. MR 1690975, DOI 10.1090/S0002-9939-00-05438-1
F. Blanchard, B. Host, and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems 20 (2000), no. 3, 641–662. MR 1764920, DOI 10.1017/S0143385700000341
Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376
Xin-Chu Fu, Yibin Fu, Jinqiao Duan, and Robert S. Mackay, Chaotic properties of subshifts generated by a nonperiodic recurrent orbit, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 (2000), no. 5, 1067–1073. MR 1770504, DOI 10.1142/S021812740000075X
Eli Glasner and Benjamin Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), no. 6, 1067–1075. MR 1251259, DOI 10.1088/0951-7715/6/6/014
Wen Huang and Xiangdong Ye, Homeomorphisms with the whole compacta being scrambled sets, Ergodic Theory Dynam. Systems 21 (2001), no. 1, 77–91. MR 1826661, DOI 10.1017/S0143385701001079
Wen Huang and Xiangdong Ye, Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl. 117 (2002), no. 3, 259–272. MR 1874089, DOI 10.1016/S0166-8641(01)00025-6
B. Kolev and M.-C. Pérouème, Recurrent surface homeomorphisms, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 1, 161–168. MR 1620528, DOI 10.1017/S0305004197002272
T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
Jiehua Mai, Continuous maps with the whole space being a scrambled set, Chinese Sci. Bull. 42 (1997), no. 19, 1603–1606. MR 1641013, DOI 10.1007/BF02882567
Jiehua Mai, Scrambled sets of continuous maps of $1$-dimensional polyhedra, Trans. Amer. Math. Soc. 351 (1999), no. 1, 353–362. MR 1473451, DOI 10.1090/S0002-9947-99-02192-3
Lex G. Oversteegen and E. D. Tymchatyn, Recurrent homeomorphisms on $\textbf {R}^2$ are periodic, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1083–1088. MR 1037216, DOI 10.1090/S0002-9939-1990-1037216-3