Strong stable manifolds for sectional-hyperbolic sets
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  to describe robustly transitive singular dynamics on $n$-manifolds (e.g. the multidimensional Lorenz attractor ). 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  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 ,  and also to the Lorenz attractor in the Lorenz equation .
Keywords: Partially Hyperbolic Set, Attractor, Flow.
Received: January 2006; Revised: August 2006; Published: December 2006.
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