Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C1 VECTOR FIELDS

  • Lee, Keonhee (Department of Mathematics Chungnam National University) ;
  • Le, Huy Tien (Department of Mathematics Vietnam National University) ;
  • Wen, Xiao (The School of Mathematics and System Science Beihang University)
  • Received : 2012.08.13
  • Published : 2014.01.01
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Abstract

Let ${\gamma}$ be a hyperbolic closed orbit of a $C^1$ vector field X on a compact boundaryless Riemannian manifold M, and let $C_X({\gamma})$ be the chain component of X which contains ${\gamma}$. We say that $C_X({\gamma})$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $\mathcal{U}$ of X such that for any $Y{\in}\mathcal{U}$, $C_Y({\gamma}_Y)$ is shadowable for $Y_t$, where ${\gamma}_Y$ denotes the continuation of ${\gamma}$ with respect to Y. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X({\gamma})$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X({\gamma})$ has no non-hyperbolic singularity.

Keywords

Acknowledgement

Supported by : National Research Foundation (NRF)

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