Strong stable manifolds for sectional-hyperbolic sets

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August 2007, 17(3): 553-560. doi: 10.3934/dcds.2007.17.553

Strong stable manifolds for sectional-hyperbolic sets

1.	Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil

Received January 2006 Revised August 2006 Published December 2006

Abstract
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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: Flow., Partially Hyperbolic Set, Attractor.

Mathematics Subject Classification: Primary: 37D30; Secondary: 37C4.

Citation: Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553

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