Some properties of coupled-expanding maps in compact sets

Zhang, Xu; Shi, Yuming; Chen, Guanrong

doi:10.1090/S0002-9939-2012-11339-5

Some properties of coupled-expanding maps in compact sets
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by Xu Zhang, Yuming Shi and Guanrong Chen PDF

Proc. Amer. Math. Soc. 141 (2013), 585-595 Request permission

Abstract:

In this paper, some properties of a strictly $A$-coupled-expanding map in compact subsets of a metric space are studied, where $A$ is a transition matrix. It is shown that this map has a compact invariant set on which it is topologically semi-conjugate to the subshift for $A$. If the subshift for $A$ has positive topological entropy, then the map is chaotic in the sense of Li-Yorke. Moreover, in the one-dimensional case, the map is at most two-to-one conjugate to the subshift for $A$ and chaotic in the sense of Devaney.

References

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
François Blanchard, Eli Glasner, Sergiĭ Kolyada, and Alejandro Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547 (2002), 51–68. MR 1900136, DOI 10.1515/crll.2002.053
L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513, DOI 10.1007/BFb0084762
Louis Block, John Guckenheimer, MichałMisiurewicz, and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. MR 591173
Alexander Blokh and Eric Teoh, How little is little enough?, Discrete Contin. Dyn. Syst. 9 (2003), no. 4, 969–978. MR 1975363, DOI 10.3934/dcds.2003.9.969
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
Gian Luigi Forti, Various notions of chaos for discrete dynamical systems. A brief survey, Aequationes Math. 70 (2005), no. 1-2, 1–13. MR 2167979, DOI 10.1007/s00010-005-2771-0
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
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
Bruce P. Kitchens, Symbolic dynamics, Universitext, Springer-Verlag, Berlin, 1998. One-sided, two-sided and countable state Markov shifts. MR 1484730, DOI 10.1007/978-3-642-58822-8
Shi Hai Li, $\omega$-chaos and topological entropy, Trans. Amer. Math. Soc. 339 (1993), no. 1, 243–249. MR 1108612, DOI 10.1090/S0002-9947-1993-1108612-8
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
Henryk Minc, Nonnegative matrices, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 932967
MichałMisiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778
Clark Robinson, Dynamical systems, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999. Stability, symbolic dynamics, and chaos. MR 1792240
S. Ruette, Chaos for Continuous Interval Maps, www.math.u-psud.fr/\verb+ +ruette/.
O. M. Šarkovs′kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. . 16 (1964), 61–71 (Russian, with English summary). MR 0159905
Yuming Shi and Guanrong Chen, Chaos of discrete dynamical systems in complete metric spaces, Chaos Solitons Fractals 22 (2004), no. 3, 555–571. MR 2062736, DOI 10.1016/j.chaos.2004.02.015
Y. Shi and G. Chen, Some new criteria of chaos induced by coupled-expanding maps, in Proc. 1st IFAC Conference on Analysis and Control of Chaotic Systems, Reims, France, June 28–30, 2006, pp. 157–162.
Yuming Shi, Hyonhui Ju, and Guanrong Chen, Coupled-expanding maps and one-sided symbolic dynamical systems, Chaos Solitons Fractals 39 (2009), no. 5, 2138–2149. MR 2519423, DOI 10.1016/j.chaos.2007.06.090
Yuming Shi and Pei Yu, Study on chaos induced by turbulent maps in noncompact sets, Chaos Solitons Fractals 28 (2006), no. 5, 1165–1180. MR 2210311, DOI 10.1016/j.chaos.2005.08.162
J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269–282. MR 849479, DOI 10.1090/S0002-9947-1986-0849479-9
Michel Vellekoop and Raoul Berglund, On intervals, transitivity = chaos, Amer. Math. Monthly 101 (1994), no. 4, 353–355. MR 1270961, DOI 10.2307/2975629
Stephen Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Texts in Applied Mathematics, vol. 2, Springer-Verlag, New York, 1990. MR 1056699, DOI 10.1007/978-1-4757-4067-7
Xiao-Song Yang and Yun Tang, Horseshoes in piecewise continuous maps, Chaos Solitons Fractals 19 (2004), no. 4, 841–845. MR 2009669, DOI 10.1016/S0960-0779(03)00202-9
Xu Zhang and Yuming Shi, Coupled-expanding maps for irreducible transition matrices, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), no. 11, 3769–3783. MR 2765092, DOI 10.1142/S0218127410028094
Z. Zhou, Symbolic Dynamics, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1997.