Specification and partial hyperbolicity for flows

Abstract

We prove that if a flow exhibits a partially hyperbolic attractor Λ with splitting and two periodic saddles with different indices such that the stable index of one of them coincides with the dimension of E^s then it does not satisfy the specification property. In particular, every sectional-hyperbolic attractor with the specification property is hyperbolic. As an application, we prove that no geometric Lorenz attractor satisfies the specification property.

Keywords:

• partially hyperbolic flows
• geometric Lorenz attractors
• specification property
• periodic points

Acknowledgements

P. Varandas was partially supported by a CNPq-Brazil postdoctoral fellowship at Universidade do Porto and N. Sumi is partially supported by KAKENHI (15K04902). P. Varandas and N. Sumi are grateful to the organizers of the conference ICM 2014 Satellite Conference on Dynamical Systems and Related Topics, Daejeon-Korea for the hospitality in Korea, where part of this work was developed. The authors are greatly indebted to the anonymous referees for their criticism and an exhaustive list of suggestions that helped to improve significantly the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

N. Sumi: KAKENHI [grant number 15K04902]; P. Varandas: CNPq-Brazil.