Saturation of generalized partially hyperbolic attractors

摘要

We prove the saturation of a generalized partially hyperbolic attractor of a map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a -generic diffeomorphism has zero volume. In contrast, by modification of the Poincare cross section of Lorenz geometric model, we build a -diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume.