Recent progress on sectional-hyperbolic systems

Abstract

We survey recent progress on the theory of sectional-hyperbolic systems. These includes sufficient conditions for sensitivity to initial conditions, essential hyperbolicity, topological properties of sectional-Anosov flows, existence of periodic and homoclinic orbits, explosion of sectional-Anosov flows, SRB measures, stochastic stability and existence of attractors.

Keywords:

• Anosov flow
• sectional-Anosov flow
• sensitive
• essentially hyperbolic
• 3-manifold

MCS/CCS/AMS Classification/CR Category numbers:

• (Primary) 37D30
• (Secondary) 37D45

Acknowledgement

C.A. Morales would like to thank Professor Keonhee Lee for his kindly invitation to the ICM 2014 Satellite Conference on Dynamical Systems and Related Topics at Chungnam National University in Daejeon, South Korea.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Announced in [46] and proved in [47].

2. Recently published in ArXiv. [49]

3. Boundary-type means that the maximal invariant set intersects just one of the two connected components of the complement of the strong stable manifold of the singularity in the stable manifold of the singularity.

Additional information

Funding

This work was partially supported by CNPq, CAPES, FAPERJ and PRONEX/Dynam. Sys. from Brazil and UNAL from Colombia.