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Journal of Modern Dynamics (JMD)
 

Tori with hyperbolic dynamics in 3-manifolds

Pages: 185 - 202, Issue 1, January 2011      doi:10.3934/jmd.2011.5.185

 
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Federico Rodriguez Hertz - IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay (email)
María Alejandra Rodriguez Hertz - IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email)
Raúl Ures - IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email)

Abstract: 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:  Anosov tori, irreducible manifold, JSJ decomposition, Seifert fibration, mapping torus, atoroidal.
Mathematics Subject Classification:  Primary: 37D20, 57M99; Secondary: 37D30, 57M50.

Received: August 2010;      Revised: February 2011;      Available Online: April 2011.

 References