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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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.