Robustly shadowable chain transitive sets and hyperbolicity

ABSTRACT

We say that a compact invariant set Λ of a C¹-vector field X on a compact boundaryless Riemannian manifold M is robustly shadowable if it is locally maximal with respect to a neighbourhood U of Λ, and there exists a C¹-neighbourhood of X such that for any , the continuation Λ_Y of Λ for Y and U is shadowable for Y_t. In this paper, we prove that any chain transitive set of a C¹-vector field on M is hyperbolic if and only if it is robustly shadowable.

KEYWORDS:

• Chain transitive sets
• dominated splitting
• hyperbolicity
• robustly shadowing

2010 Mathematics Subject Classification Codes:

• 37D40
• 37C50

Acknowledgments

The authors would like to thank the referees for reading of the manuscript and useful comments. The second author was supported by the NRF grant funded by the Korea government (MSIP) (No. NRF-2015R1A2A2A01002437).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

NRF grant funded by the Korea government (Ministry of Science, ICT and Future Planning (MSIP)) [grant number 2015R1A2A2A01002437].