Ergodic Theory and Dynamical Systems

Research Article

Sectional-hyperbolic systems

R. METZGERa1 and C. MORALESa2

a1 Instituto de Matemática y Ciencias Afines IMCA, Jr. Ancash 536. Lima 1., Casa de las Trece Monedas, Perú (email: metzger@uni.edu.pe)

a2 Instituto de Matematica, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970 Rio de Janeiro, Brazil (email: morales@impa.br)

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].

(Received August 05 2005)

(Revised November 06 2007)