Surgery for partially hyperbolic dynamical systems, I: Blow-ups of invariant submanifolds

We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism $f : M \to M$ which leaves invariant a submanifold $N \subset M$ . We assume that $N$ is an Anosov submanifold for $f$ , that is, the restriction $f |_{N}$ is an Anosov diffeomorphism and the center distribution is transverse to $T N \subset T M$ . By replacing each point in $N$ with the projective space (real or complex) of lines normal to $N$ , we obtain the blow-up $\hat{M}$ . Replacing $M$ with $\hat{M}$ amounts to a surgery on the neighborhood of $N$ which alters the topology of the manifold. The diffeomorphism $f$ induces a canonical diffeomorphism $\hat{f} : \hat{M} \to \hat{M}$ . We prove that under certain assumptions on the local dynamics of $f$ at $N$ the diffeomorphism $\hat{f}$ is also partially hyperbolic. We also present some modifications, such as the connected sum construction, which allows to “paste together” two partially hyperbolic diffeomorphisms to obtain a new one. Finally, we present several examples to which our results apply.

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Keywords

partially hyperbolic diffeomorphism, surgery, blow-up, Anosov submanifold, fiberwise Anosov diffeomorphism

Mathematical Subject Classification 2010

Primary: 37D30

References

Publication

Received: 20 September 2016

Accepted: 1 August 2017

Published: 5 April 2018

Proposed: Dmitri Burago

Seconded: Leonid Polterovich, Bruce Kleiner

Authors

Andrey Gogolev
Department of Mathematics Ohio State University Columbus, OH United States	Department of Mathematical Sciences Binghamton University, State University of New York Binghamton, NY United States

Volume 22, issue 4 (2018)

Andrey Gogolev

Abstract