Singular-hyperbolic sets and topological dimension
Title:
Singular-hyperbolic sets and topological dimension
Author:
Morales, C. A.
Appeared in:
Dynamical systems
Paging:
Volume 18 (2003) nr. 2 pages 181-189
Year:
2003-06-01
Contents:
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).
Journal description