Hyperbolicity, transitivity and the two-sided limit shadowing property
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2013
- DOI:
- 10.48550/arXiv.1301.2356
- arXiv:
- arXiv:1301.2356
- Bibcode:
- 2013arXiv1301.2356C
- Keywords:
-
- Mathematics - Dynamical Systems;
- 37D20;
- 37C20
- E-Print:
- 10 pages