Singular-hyperbolic sets and topological dimension

摘要

A 'singular-hyperbolic set' for flows is a partially hyperbolic set with singularities (hyperbolic ones) and volume expanding central direction (Morales et al. 1998, Comptes Rendus de L' Academic des Sciences Paris-Serie I―Mathematics, 326: 81-86). The class of transitive singular-hyperbolic sets includes the geometric Lorenz attractor and the singular horseshoe (Guckenheimer and Williams 1979, inst. Hautes Etudes Sci Publ. Math., 50: 59―72, Labarca and Pacifico 1986, Topology, 25: 337―352). We prove that all compact, non-singular, invariant subsets of a transitive singular-hyperbolic set are 1-dimensional. This generalizes the minimal set's results for Axiom A flows in Bowen (1973, American Journal of Mathematics, 95: 429-460) to a class of flows studied in Morales et al. (1999, Proceedings of the American Mathematical Society, 127: 3393―3401).