Journal of Modern Dynamics (JMD)

Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups

Pages: 541 - 580, Issue 4, October 2008      doi:10.3934/jmd.2008.2.541

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Dmitri Burago - Department ofMathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)
Sergei Ivanov - Steklov Math. Institute, 27, Fontanka, St. Petersburg 191023, Russian Federation (email)

Abstract: We present the first known nontrivial topological obstructions to the existence of partially hyperbolic diffeomorphisms. In particular, we show that there are no partially hyperbolic diffeomorphisms on the 3-sphere. More generally, we show that for a partially hyperbolic diffeomorphism of a 3-mani-fold with an Abelian fundamental group, the induced action in the first homology group is partially hyperbolic. This improves the results of [4] by dropping the assumption of dynamical coherence.

Keywords:  partially hyperbolic diffeomorphism; dynamical coherence; stable, unstable, center distributions and foliations; quasi-isometric leaves
Mathematics Subject Classification:  34C40, 34D45, 37C05, 37C70, 37D10, 37D30

Received: February 2007;      Revised: September 2008;      Available Online: October 2008.