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C 1 Interiors of Sets of Systems with Various Shadowing Properties

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Shadowing and Hyperbolicity

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2193))

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Abstract

In this chapter, we study the structure of C 1 interiors of some basic sets of dynamical systems having various shadowing properties. We give either complete proofs or schemes of proof of the following main results:

  • The C 1 interior of the set of diffeomorphisms having the standard shadowing property is a subset of the set of structurally stable diffeomorphisms (Theorem 3.1.1); this result and Theorem 1.4.1 (a) imply that the C 1 interior of the set of diffeomorphisms having the standard shadowing property coincides with the set of structurally stable diffeomorphisms;

  • the set \(\mbox{ Int}^{1}(\mbox{ OrientSP}_{F}\setminus \mathcal{B})\) is a subset of the set of structurally stable vector fields (Theorem 3.3.1); similarly to the case of diffeomorphisms, this result and Theorem 1.4.1 (b) imply that the set \(\mbox{ Int}^{1}(\mbox{ OrientSP}_{F}\setminus \mathcal{B})\) coincides with the set of structurally stable vector fields;

  • the set Int1(OrientSP F ) contains vector fields that are not structurally stable (Theorem 3.4.1).

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Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Sergei Yu. Pilyugin

  2. Faculty of Education, Utsunomiya University, Utsunomiya, Japan

    Kazuhiro Sakai

Authors
  1. Sergei Yu. Pilyugin
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  2. Kazuhiro Sakai
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Pilyugin, S.Y., Sakai, K. (2017). C 1 Interiors of Sets of Systems with Various Shadowing Properties. In: Shadowing and Hyperbolicity. Lecture Notes in Mathematics, vol 2193. Springer, Cham. https://doi.org/10.1007/978-3-319-65184-2_3

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