Quasi-Anosov diffeomorphisms of 3-manifolds

Fisher, T.; Rodriguez Hertz, M.

doi:10.1090/S0002-9947-09-04687-X

Quasi-Anosov diffeomorphisms of 3-manifolds
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Trans. Amer. Math. Soc. 361 (2009), 3707-3720 Request permission

Abstract:

In 1969, Hirsch posed the following problem: given a diffeomorphism $f:N\to N$ and a compact invariant hyperbolic set $\Lambda$ of $f$, describe the topology of $\Lambda$ and the dynamics of $f$ restricted to $\Lambda$. We solve the problem where $\Lambda =M^3$ is a closed $3$-manifold: if $M^3$ is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions.

The dynamics of the diffeomorphisms restricted to $M^3$, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions.

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