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Sectional-hyperbolic systems

Published online by Cambridge University Press:  01 October 2008

R. METZGER  and
C. MORALES
R. METZGER
Affiliation:
Instituto de Matemática y Ciencias Afines IMCA, Jr. Ancash 536. Lima 1., Casa de las Trece Monedas, Perú (email: [email protected])
C. MORALES
Affiliation:
Instituto de Matematica, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970 Rio de Janeiro, Brazil (email: [email protected])
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Abstract

We introduce a class of vector fields on n-manifolds containing the hyperbolic systems, the singular-hyperbolic systems on 3-manifolds, the multidimensional Lorenz attractors and the robust transitive singular sets in Li et al [Robust transitive singular sets via approach of an extended linear Poincaré flow. Discrete Contin. Dyn. Syst.13(2) (2005), 239–269]. We prove that the closed orbits of a system in such a class are hyperbolic in a persistent way, a property which is false for higher-dimensional singular-hyperbolic systems. We also prove that the singularities in the robust transitive sets in Li et al are similar to those in the multidimensional Lorenz attractor. Our results will give a partial negative answer to Problem 9.26 in Bonatti et al [Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1587 - 1597
Copyright
Copyright © 2008 Cambridge University Press

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