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Hyperbolicity, transitivity and the two-sided limit shadowing property


Author: Bernardo Carvalho
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
Published electronically: October 3, 2014
MathSciNet review: 3283652
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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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Bernardo Carvalho
Affiliation: Departamento de Matemática Pura, Universidade Federal do Rio de Janeiro - UFRJ, Cidade Universitária, Rio de Janeiro - RJ, 21941-901, Brazil
Email: bmcarvalho06@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12250-7
Keywords: Pseudo-orbit, shadowing, limit shadowing, hyperbolicity, transitivity
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
Article copyright: © Copyright 2014 American Mathematical Society