Hyperbolicity, transitivity and the two-sided limit shadowing property
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2013
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| _version_ | 1866917809173299200 |
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| author | Carvalho, Bernardo |
| author_facet | Carvalho, Bernardo |
| contents | We explore the notion of two-sided limit shadowing property introduced by Pilyugin \cite{P1}. 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 codimention-one Anosov diffeomorphisms have the two-sided limit shadowing property. We also prove that every diffeomorphism $f$ with such a property on a closed manifold has neither sinks nor sources and is transitive Anosov (in the Axiom A case). In particular, no Morse-Smale diffeomorphism have the two-sided limit shadowing property. Finally, we prove that $C^1$-generic diffeomorphisms on closed manifolds 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 on closed manifolds to prove that the set of diffeomorphisms with the two-sided limit shadowing property coincides with the set of Anosov diffeomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1301_2356 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Hyperbolicity, transitivity and the two-sided limit shadowing property Carvalho, Bernardo Dynamical Systems 37D20, 37C20 We explore the notion of two-sided limit shadowing property introduced by Pilyugin \cite{P1}. 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 codimention-one Anosov diffeomorphisms have the two-sided limit shadowing property. We also prove that every diffeomorphism $f$ with such a property on a closed manifold has neither sinks nor sources and is transitive Anosov (in the Axiom A case). In particular, no Morse-Smale diffeomorphism have the two-sided limit shadowing property. Finally, we prove that $C^1$-generic diffeomorphisms on closed manifolds 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 on closed manifolds to prove that the set of diffeomorphisms with the two-sided limit shadowing property coincides with the set of Anosov diffeomorphisms. |
| title | Hyperbolicity, transitivity and the two-sided limit shadowing property |
| topic | Dynamical Systems 37D20, 37C20 |
| url | https://arxiv.org/abs/1301.2356 |