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DCDS

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.

JMD

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.
JMD

Let $M$ be a closed orientable irreducible $3$-dimensional manifold. An
embedded $2$-torus $\mathbb{T}$ is an

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].

*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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