## Hyperbolicity, transitivity and the two-sided limit shadowing property

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**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.## References

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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