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October  2008, 2(4): 541-580. doi: 10.3934/jmd.2008.2.541

Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups

Dmitri Burago 1, and  Sergei Ivanov 2,

1. 

Department ofMathematics, The Pennsylvania State University, University Park, PA 16802, United States
2. 

Steklov Math. Institute, 27, Fontanka, St. Petersburg 191023, Russian Federation

Received  February 2007 Revised  September 2008 Published  October 2008

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: unstable, partially hyperbolic diffeomorphism; dynamical coherence; stable, center distributions and foliations; quasi-isometric leaves.
Mathematics Subject Classification: 34C40, 34D45, 37C05, 37C70, 37D10, 37D30.
Citation: Dmitri Burago, Sergei Ivanov. Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups. Journal of Modern Dynamics, 2008, 2 (4) : 541-580. doi: 10.3934/jmd.2008.2.541
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