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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hyperbolicity, transitivity and the two-sided limit shadowing property
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by Bernardo Carvalho PDF
Proc. Amer. Math. Soc. 143 (2015), 657-666 Request permission

Abstract:

We explore the notion of the two-sided limit shadowing property introduced by Pilyugin in 2007. Indeed, we characterize the $C^1$-interior of the set of diffeomorphisms with such a property on closed manifolds as the set of transitive Anosov diffeomorphisms. As a consequence we obtain that all codimension-one Anosov diffeomorphisms have the two-sided limit shadowing property. We also prove that every diffeomorphism $f$ with such a property has neither sinks nor sources and is transitive Anosov (in the Axiom A case). In particular, no Morse-Smale diffeomorphism has the two-sided limit shadowing property. Finally, we prove that $C^1$-generic diffeomorphisms with the two-sided limit shadowing property are transitive Anosov. All these results allow us to reduce the well-known conjecture about the transitivity of Anosov diffeomorphisms to prove that the set of diffeomorphisms with the two-sided limit shadowing property coincides with the set of Anosov diffeomorphisms.
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Additional Information
  • Bernardo Carvalho
  • Affiliation: Departamento de Matemática Pura, Universidade Federal do Rio de Janeiro - UFRJ, Cidade Universitária, Rio de Janeiro - RJ, 21941-901, Brazil
  • MR Author ID: 1027591
  • ORCID: 0000-0002-9400-0882
  • Email: bmcarvalho06@gmail.com
  • Received by editor(s): January 10, 2013
  • Received by editor(s) in revised form: April 26, 2013
  • Published electronically: October 3, 2014
  • Additional Notes: This paper was partially supported by CAPES (Brazil)
  • Communicated by: Yingfei Yi
  • © Copyright 2014 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 657-666
  • MSC (2010): Primary 37D20; Secondary 37C20
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12250-7
  • MathSciNet review: 3283652