ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C¹ VECTOR FIELDS
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Let γ be a hyperbolic closed orbit of a C¹ vector field X on a compact boundaryless Riemannian manifold M, and let Cχ(γ) be the chain component of X which contains γ. We say that Cχ(γ) is C¹ ro- bustly shadowable if there is a C¹ neighborhood U of X such that for any Y ∈ U, Cr(γr ) is shadowable for Y₁, where γr denotes the continuation of γ with respect to Y. In this paper, we prove that any C¹ robustly shad- owable chain component Cχ(γ) does not contain a hyperbolic singularity, and it is hyperbolic if Cχ(γ) has no non-hyperbolic singularity.
Abstract
Let γ be a hyperbolic closed orbit of a C¹ vector field X on a compact boundaryless Riemannian manifold M, and let Cχ(γ) be the chain component of X which contains γ. We say that Cχ(γ) is C¹ ro- bustly shadowable if there is a C¹ neighborhood U of X such that for any Y ∈ U, Cr(γr ) is shadowable for Y₁, where γr denotes the continuation of γ with respect to Y. In this paper, we prove that any C¹ robustly shad- owable chain component Cχ(γ) does not contain a hyperbolic singularity, and it is hyperbolic if Cχ(γ) has no non-hyperbolic singularity.
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