Chain transitive sets and shadowing

Sergei Yu Pilyugin; Kazuhiro Sakai

doi:10.1007/978-3-319-65184-2_4

Chain transitive sets and shadowing

Sergei Yu Pilyugin, Kazuhiro Sakai

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

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Abstract

In this chapter, we study relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets. We prove the following two main results: • Let ⋀ be a closed invariant set of f ϵ Diff¹(M). Then f|_⋀ is chain transitive and C¹-stably shadowing in a neighborhood of ⋀ if and only if ⋀ is a hyperbolic basic set (Theorem 4.2.1); • there is a residual set R ⊂ Diff¹(M) such that if f ϵ R and ⋀ is a locally maximal chain transitive set of f, then ⋀ is hyperbolic if and only if f |_⋀ is shadowing (Theorem 4.3.1).

Original language	English
Title of host publication	Lecture Notes in Mathematics
Publisher	Springer Nature
Pages	181-208
Number of pages	28
DOIs	https://doi.org/10.1007/978-3-319-65184-2_4
State	Published - 2017

Publication series

Name	Lecture Notes in Mathematics
Volume	2193
ISSN (Print)	0075-8434

Scopus subject areas

Algebra and Number Theory

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10.1007/978-3-319-65184-2_4

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abstract = "In this chapter, we study relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets. We prove the following two main results: • Let ⋀ be a closed invariant set of f ϵ Diff1(M). Then f|⋀ is chain transitive and C1-stably shadowing in a neighborhood of ⋀ if and only if ⋀ is a hyperbolic basic set (Theorem 4.2.1); • there is a residual set R ⊂ Diff1(M) such that if f ϵ R and ⋀ is a locally maximal chain transitive set of f, then ⋀ is hyperbolic if and only if f |⋀ is shadowing (Theorem 4.3.1).",

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AB - In this chapter, we study relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets. We prove the following two main results: • Let ⋀ be a closed invariant set of f ϵ Diff1(M). Then f|⋀ is chain transitive and C1-stably shadowing in a neighborhood of ⋀ if and only if ⋀ is a hyperbolic basic set (Theorem 4.2.1); • there is a residual set R ⊂ Diff1(M) such that if f ϵ R and ⋀ is a locally maximal chain transitive set of f, then ⋀ is hyperbolic if and only if f |⋀ is shadowing (Theorem 4.3.1).

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Chain transitive sets and shadowing

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