DCDS
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center
Keith Burns Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Anna Talitskaya Raúl Ures
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 75-88 doi: 10.3934/dcds.2008.22.75
It is shown that stable accessibility property is $C^r$-dense among partially hyperbolic diffeomorphisms with one-dimensional center bundle, for $r \geq 2$, volume preserving or not. This establishes a conjecture by Pugh and Shub for these systems.
keywords: density of accessibility one dimensional center. partial hyperbolicity
JMD
Partial hyperbolicity and ergodicity in dimension three
Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Raúl Ures
Journal of Modern Dynamics 2008, 2(2): 187-208 doi: 10.3934/jmd.2008.2.187
In [15] the authors proved the Pugh–Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e., stably ergodic diffeomorphisms are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the first examples of manifolds in which all conservative partially hyperbolic diffeomorphisms are ergodic.
keywords: partial hyperbolicity laminations. accessibility property ergodicity
JMD
Tori with hyperbolic dynamics in 3-manifolds
Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Raúl Ures
Journal of Modern Dynamics 2011, 5(1): 185-202 doi: 10.3934/jmd.2011.5.185
Let $M$ be a closed orientable irreducible $3$-dimensional manifold. An embedded $2$-torus $\mathbb{T}$ is an Anosov torus if there exists a diffeomorphism $f$ over $M$ for which $\T$ is $f$-invariant and $f_\#|_\mathbb{T}:\pi_1(\mathbb{T})\to \pi_1(\mathbb{T})$ is hyperbolic. We prove that only few irreducible $3$-manifolds admit Anosov tori: (1) the $3$-torus $\mathbb{T}^3$; (2) the mapping torus of $-\Id$; and (3) the mapping tori of hyperbolic automorphisms of $\mathbb{T}^2$.
   This has consequences for instance in the context of partially hyperbolic dynamics of $3$-manifolds: if there is an invariant foliation $\mathcal{F}^{cu}$ tangent to the center-unstable bundle $E^c\oplus E^u$, then $\mathcal{F}^{cu}$ has no compact leaves [21]. This has led to the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle [21].
keywords: irreducible manifold mapping torus JSJ decomposition Anosov tori atoroidal. Seifert fibration

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