Saturation of generalized partially hyperbolic attractors

ABSTRACT

We prove the saturation of a generalized partially hyperbolic attractor of a $C^2$ map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a $C^1$-generic diffeomorphism has zero volume. In contrast, by modification of the Poincaré cross section of Lorenz geometric model, we build a $C^1$-diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume.

KEYWORDS:

• Generalized partially hyperbolic attractor
• $C^1$-generic
• horseshoe-like
• Lorenz attractor

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37DXX
• 37XX

Acknowledgments

During the prepration of this article, the first author was partially supported by grant from IPM (No. 96370119). He also thanks ICTP for supporting through the association schedule.

Disclosure statement>

No potential conflict of interest was reported by the authors.

Notes

1. We call a horseshoe H fat if ${\mathrm{Leb}}_2\left(H\right)>0$.

2. The attractor contains the unstable manifold of the hyperbolic equilibrium point.

Additional information

Funding

During the preparation of this article the first author was partially supported by grant from IPM [No. 96370119].