Strong stable manifolds for sectional-hyperbolic sets

Abstract

The <i> sectional-hyperbolic sets</i> 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 <i> singular-hyperbolic attractor</i> with singularities [22] on a compact $3$-manifold has topological dimension $2$. Hence such an attractor is <i> expanding</i>, 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].