Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Strong stable manifolds for sectional-hyperbolic sets

Pages: 553 - 560, Volume 17, Issue 3, March 2007

doi:10.3934/dcds.2007.17.553       Abstract        Full Text (145.6K)       Related Articles

Carlos Arnoldo Morales - Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil (email)

Abstract: The sectional-hyperbolic sets constitute a class of partially hyperbolic sets introduced in [20] to describe robustly transitive singular dynamics on $n$-manifolds (e.g. the multidimensional Lorenz attractor [9]). Here we prove that a transitive sectional-hyperbolic set with singularities contains no local strong stable manifold through any of its points. Hence a transitive, isolated, sectional-hyperbolic set containing a local strong stable manifold is a hyperbolic saddle-type repeller. In addition, a proper transitive sectional-hyperbolic set on a compact $n$-manifold has empty interior and topological dimension $\leq n-1$. It follows that a singular-hyperbolic attractor with singularities [22] on a compact $3$-manifold has topological dimension $2$. Hence such an attractor is expanding, i.e., its topological dimension coincides with the dimension of its central subbundle. These results apply to the robustly transitive sets considered in [22], [17] and also to the Lorenz attractor in the Lorenz equation [25].

Keywords:  Partially Hyperbolic Set, Attractor, Flow.
Mathematics Subject Classification:  Primary: 37D30; Secondary: 37C45.

Received: January 2006;      Revised: August 2006;      Published: December 2006.