PARTIALLY HYPERBOLIC SETS WITH POSITIVE MEASURE AND ACIP FOR PARTIALLY HYPERBOLIC SYSTEMS

摘要

In [23] Xia introduced a simple dynamical density basis for partially hyperbolic sets of volume preserving diffeomorphisms. We apply the density basis to the study of the topological structure of partially hyperbolic sets. We show that if A is a strongly partially hyperbolic set with positive volume, then A contains the global stable manifolds over α(Λ~d) and the global unstable manifolds over ω(Λ~d). We give several applications of the dynamical density to partially hyperbolic maps that preserve some acip. We show that if / is essentially accessible and μ is an acip of f, then supp(μ) = M, the map / is transitive, and μ-a.e. x ∈ M has a dense orbit in M. Moreover if f is accessible and center bunched, then either f preserves a smooth measure or there is no acip at all.