On the volume of singular-hyperbolic sets

• Type: Article in International Scientific Journal
• Date: 2007

Authors

alves, jf (Author) FCUP; arauo, v (Author) Other; pacifico, mj (Author) Other; pinheiro, v (Author) Other

Journal

Title: Dynamical Systems

Vol. 22

Pages: 249-267

ISSN: 1468-9367

Publisher: Taylor & Francis

Indexing

ISI Web of Knowledge - 10 Citations

Scopus - 12 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-004-CYY

DOI: 10.1080/14689360601165217

Abstract (EN): An attractor Lambda for a 3-vector field X is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that C1+alpha singular-hyperbolic attractors, for any alpha > 0, always have zero volume, extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular attractors. Moreover, we prove that if Lambda is a singular-hyperbolic attractor for X then either it has zero volume or X is an Anosov flow. We also present examples of C-1 singular-hyperbolic attractors with positive volume. In addition, we show that C-1 generically we have volume zero for C-1 robust classes of singular-hyperbolic attractors.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 19