5,897 research outputs found

    Homoclinic intersections of symplectic partially hyperbolic systems with 2D center

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    We study some generic properties of partially hyperbolic symplectic systems with 2D center. We prove that CrC^r generically, every hyperbolic periodic point has a transverse homoclinic intersection for the maps close to a direct/skew product of an Anosov diffeomorphism with a map on S2S^2 or T2\mathbb{T}^2

    A diffeomorphism with global dominated splitting can not be minimal

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    Let M be a closed manifold and f be a diffeomorphism on M. We show that if f has a nontrivial dominated splitting TM=E\oplus F, then f can not be minimal. The proof mainly use Mane's argument and Liao's selecting lemma.Comment: 5 pages. An application of Liao's selecting lemm

    Partially hyperbolic sets with positive measure and ACIPACIP for partially hyperbolic systems

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    In [Discrete Contin. Dyn. Syst. \textbf{15} (2006), no. 3, 811--818.] 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 Λ\Lambda is a strongly partially hyperbolic set with positive volume, then Λ\Lambda contains the global stable manifolds over α(Λd){\alpha}(\Lambda^d) and the global unstable manifolds over ω(Λd){\omega}(\Lambda^d). We give several applications of the dynamical density to partially hyperbolic maps that preserve some acipacip. We show that if ff is essentially accessible and μ\mu is an acipacip of ff, then supp(μ)=M\text{supp}(\mu)=M, the map ff is transitive, and μ\mu-a.e. x∈Mx\in M has a dense orbit in MM. Moreover if ff is accessible and center bunched, then either ff preserves a smooth measure or there is no acipacip of ff.Comment: Correct the proof of Theorem 5.5. Add a few explanation
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