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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Quasi-Anosov diffeomorphisms of 3-manifolds


Authors: T. Fisher and M. Rodriguez Hertz
Journal: Trans. Amer. Math. Soc. 361 (2009), 3707-3720
MSC (2000): Primary 37D05, 37D20
Published electronically: February 10, 2009
MathSciNet review: 2491896
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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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Additional Information

T. Fisher
Affiliation: Department of Mathematics, Brigham Young University, 292 TMCB, Provo, Utah 84602
Email: tfisher@math.byu.edu

M. Rodriguez Hertz
Affiliation: IMERL, Facultad de Ingeniería, University de la Republica, Julio Herrera y Reissig 565, 11300 Montevideo, Uruguay
Email: jana@fing.edu.uy

DOI: https://doi.org/10.1090/S0002-9947-09-04687-X
Keywords: Dynamical systems, hyperbolic set, robustly expansive, quasi-Anosov
Received by editor(s): May 8, 2007
Published electronically: February 10, 2009
Additional Notes: This work was partially supported by NSF Grant #DMS0240049, Fondo Clemente Estable 9021 and PDT
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.