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

Andrey Gogolev

Geometry & Topology 22 (2018) 2219–2252

We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism f : M M which leaves invariant a submanifold N 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 TN TM. By replacing each point in N with the projective space (real or complex) of lines normal to N, we obtain the blow-up M̂. Replacing M with M̂ amounts to a surgery on the neighborhood of N which alters the topology of the manifold. The diffeomorphism f induces a canonical diffeomorphism f̂: M̂ M̂. We prove that under certain assumptions on the local dynamics of f at N the diffeomorphism f̂ 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.

partially hyperbolic diffeomorphism, surgery, blow-up, Anosov submanifold, fiberwise Anosov diffeomorphism
Mathematical Subject Classification 2010
Primary: 37D30
Received: 20 September 2016
Accepted: 1 August 2017
Published: 5 April 2018
Proposed: Dmitri Burago
Seconded: Leonid Polterovich, Bruce Kleiner
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