Chain transitive sets and shadowing

Sergei Yu Pilyugin, Kazuhiro Sakai

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

1 Scopus citations

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 languageEnglish
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Nature
Pages181-208
Number of pages28
DOIs
StatePublished - 2017

Publication series

NameLecture Notes in Mathematics
Volume2193
ISSN (Print)0075-8434

Scopus subject areas

  • Algebra and Number Theory

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