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Coupled-expanding maps and one-sided symbolic dynamical systems

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  • Shi, Yuming
  • Ju, Hyonhui
  • Chen, Guanrong

Abstract

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.

Suggested Citation

  • Shi, Yuming & Ju, Hyonhui & Chen, Guanrong, 2009. "Coupled-expanding maps and one-sided symbolic dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2138-2149.
  • Handle: RePEc:eee:chsofr:v:39:y:2009:i:5:p:2138-2149
    DOI: 10.1016/j.chaos.2007.06.090
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    References listed on IDEAS

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    1. Shi, Yuming & Yu, Pei, 2006. "Study on chaos induced by turbulent maps in noncompact sets," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1165-1180.
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    Cited by:

    1. Ju, Hyonhui & Kim, Cholsan & Choe, Yunmi & Chen, Minghao, 2017. "Conditions for topologically semi-conjugacy of the induced systems to the subshift of finite type," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 1-6.
    2. Kim, Cholsan & Ju, Hyonhui & Chen, Minghao & Raith, Peter, 2015. "A-coupled-expanding and distributional chaos," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 291-295.
    3. Kim, Jinhyon & Ju, Hyonhui, 2018. "Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 246-251.

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