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Coupled-expanding maps and one-sided symbolic dynamical systems. (English) Zbl 1197.37010

Summary: This paper studies relationships between coupled-expanding maps and one-sided symbolic dynamical systems. The concept of coupled-expanding map is extended to a more general one: coupled-expansion for a transitive matrix. It is found that the subshift for a transitive matrix is strictly coupled-expanding for the matrix in certain disjoint compact subsets; the topological conjugacy of a continuous map in its compact invariant set of a metric space to a subshift for a transitive matrix has a close relationship with that the map is strictly coupled-expanding for the matrix in some disjoint compact subsets. A certain relationship between strictly coupled-expanding maps for a transitive matrix in disjoint bounded and closed subsets of a complete metric space and their topological conjugacy to the subshift for the matrix is also obtained. Dynamical behaviors of subshifts for irreducible matrices are then studied and several equivalent statements to chaos are obtained; especially, chaos in the sense of Li-Yorke is equivalent to chaos in the sense of Devaney for the subshift, and is also equivalent to that the domain of the subshift is infinite. Based on these results, several new criteria of chaos for maps are finally established via strict coupled-expansions for irreducible transitive matrices in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, respectively, where their conditions are weaker than those existing in the literature.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

37B10 Symbolic dynamics
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[1] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On devaney’s definition of chaos, Amer math monthly, 99, 332-334, (1992) · Zbl 0758.58019
[2] Birkhoff GD. Dynamical systems. Providence: AMS Publications; 1927. Reprinted; 1966.
[3] Birkhoff, G.D., Nouvelles recherches sur LES systèmes dynamiques, Mem point acad sci novi lyncaei, 1, 85-216, (1935) · Zbl 0016.23401
[4] Block, L.S.; Coppel, W.A., Dynamics in one dimension, Lecture notes in mathematics, vol. 1513, (1992), Springer-Verlag Berlin/Heidelberg
[5] Devaney, R.L., An introduction to chaotic dynamical systems, (1989), Addison-Wesley Publishing Company · Zbl 0695.58002
[6] Devaney, R.L.; Nitecki, Z., Shift automorphism in the Hénon mapping, Commun math phys, 67, 137-148, (1979) · Zbl 0414.58028
[7] Hadamard, J., LES surfaces à curbures opposés et leurs lignes géodesiques, J math, 5, 27-73, (1898) · JFM 29.0522.01
[8] Hao, B.L.; Zheng, W.M., Applied symbolic dynamics and chaos, (1998), World Scientific Publishing Company · Zbl 0914.58017
[9] Huang, W.; Ye, X., Devaney’s chaos or 2-scattering implies li – yorke’s chaos, Topol appl, 117, 259-272, (2002) · Zbl 0997.54061
[10] Kennedy, J.; Yorke, J.A., Topological horseshoes, Trans amer math soc, 353, 2513-2530, (2001) · Zbl 0972.37011
[11] Li, T.; Yorke, J.A., Period three implies chaos, Amer math monthly, 82, 985-992, (1975) · Zbl 0351.92021
[12] Marotto, F.R., Snap-back repellers imply chaos in rn, J math anal appl, 63, 199-223, (1978) · Zbl 0381.58004
[13] Morse, M.; Hedlund, G.A., Symbolic dynamics, Amer J math, 60, 815-866, (1938) · JFM 64.0798.04
[14] Moser, J., Stable and random motions in dynamical systems, (1973), Princeton University Press Princeton
[15] Robinson, C., Dynamical systems: stability, symbolic dynamics and chaos, (1995), CRC Press FL · Zbl 0853.58001
[16] Shi, Y.; Chen, G., Chaos of discrete dynamical systems in complete metric spaces, Chaos, solitons & fractals, 22, 555-571, (2004) · Zbl 1067.37047
[17] Shi Y, Chen G. Discrete chaos in Banach spaces. Science in China Ser A: Mathematics, Chinese version 2004; 34: 595-609. English version 2005; 48: p. 222-38.
[18] Shi Y, Chen G. Some new criteria of chaos induced by coupled-expanding maps. In: Proc. the 1st IFAC conference on analysis and control of chaotic systems, Reims, France, June 28-30, 2006, p. 157-62.
[19] Shi, Y.; Yu, P., Study on chaos induced by turbulent maps in noncompact sets, Chaos, solitons & fractals, 28, 1165-1180, (2006) · Zbl 1106.37008
[20] Shi, Y.; Yu, P., Chaos induced by regular snap-back repellers, J math anal appl., 337, 1480-1494, (2008) · Zbl 1131.37023
[21] Smale, S., Diffeomorphisms with many periodic points, (), 63-80 · Zbl 0142.41103
[22] Smale, S., Differentiable dynamical systems, Bull amer math soc, 73, 747-817, (1967) · Zbl 0202.55202
[23] Wiggins, S., Global bifurcation and chaos, (1988), Springer-Verlag New York
[24] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (1990), Springer-Verlag New York · Zbl 0701.58001
[25] Zhou, Z., Symbolic dynamics, (1997), Shanghai Scientific and Technological Education Publishing House Shanghai, [in Chinese]
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.