Topological structure of (partially) hyperbolic sets with positive volume

Authors:
José F. Alves and Vilton Pinheiro

Journal:
Trans. Amer. Math. Soc. **360** (2008), 5551-5569

MSC (2000):
Primary 37Dxx

DOI:
https://doi.org/10.1090/S0002-9947-08-04484-X

Published electronically:
April 28, 2008

MathSciNet review:
2415085

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Abstract: We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is larger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also give a description of the limit set of almost every point belonging to a hyperbolic set or a partially hyperbolic set with positive volume.

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Additional Information

**José F. Alves**

Affiliation:
Departamento de Matemática Pura, Faculdade de Ciências do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Email:
jfalves@fc.up.pt

**Vilton Pinheiro**

Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil

Email:
viltonj@ufba.br

DOI:
https://doi.org/10.1090/S0002-9947-08-04484-X

Keywords:
Hyperbolic set,
partially hyperbolic set,
horseshoe

Received by editor(s):
June 5, 2006

Received by editor(s) in revised form:
January 8, 2007

Published electronically:
April 28, 2008

Additional Notes:
This work was carried out at the Federal University of Bahia, University of Porto and IMPA. The first author was partially supported by CMUP, by a grant of FCT and by POCI/MAT/61237/2004. The second author was partially supported by PADCT/CNPq and by POCI/MAT/61237/2004

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.