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Chain transitive sets and shadowing. / Pilyugin, Sergei Yu; Sakai, Kazuhiro.

Lecture Notes in Mathematics. Springer Nature, 2017. p. 181-208 (Lecture Notes in Mathematics; Vol. 2193).

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Harvard

Pilyugin, SY & Sakai, K 2017, Chain transitive sets and shadowing. in Lecture Notes in Mathematics. Lecture Notes in Mathematics, vol. 2193, Springer Nature, pp. 181-208. https://doi.org/10.1007/978-3-319-65184-2_4

APA

Pilyugin, S. Y., & Sakai, K. (2017). Chain transitive sets and shadowing. In Lecture Notes in Mathematics (pp. 181-208). (Lecture Notes in Mathematics; Vol. 2193). Springer Nature. https://doi.org/10.1007/978-3-319-65184-2_4

Vancouver

Pilyugin SY, Sakai K. Chain transitive sets and shadowing. In Lecture Notes in Mathematics. Springer Nature. 2017. p. 181-208. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-319-65184-2_4

Author

Pilyugin, Sergei Yu ; Sakai, Kazuhiro. / Chain transitive sets and shadowing. Lecture Notes in Mathematics. Springer Nature, 2017. pp. 181-208 (Lecture Notes in Mathematics).

BibTeX

@inbook{28e80dab1d7b4c00bc9da39c9cda75fe,

title = "Chain transitive sets and shadowing",

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).",

author = "Pilyugin, {Sergei Yu} and Kazuhiro Sakai",

year = "2017",

doi = "10.1007/978-3-319-65184-2_4",

language = "English",

series = "Lecture Notes in Mathematics",

publisher = "Springer Nature",

pages = "181--208",

booktitle = "Lecture Notes in Mathematics",

address = "Germany",

}

RIS

TY - CHAP

T1 - Chain transitive sets and shadowing

AU - Pilyugin, Sergei Yu

AU - Sakai, Kazuhiro

PY - 2017

Y1 - 2017

N2 - 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).

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).

UR - http://www.scopus.com/inward/record.url?scp=85029091668&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-65184-2_4

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

M3 - Chapter

AN - SCOPUS:85029091668

T3 - Lecture Notes in Mathematics

SP - 181

EP - 208

BT - Lecture Notes in Mathematics

PB - Springer Nature

ER -

ID: 74985808