#### Volume 22, issue 4 (2018)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Surgery for partially hyperbolic dynamical systems, I: Blow-ups of invariant submanifolds

### Andrey Gogolev

Geometry & Topology 22 (2018) 2219–2252
##### Abstract

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 $N\subset M\phantom{\rule{0.3em}{0ex}}$. 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 $TN\subset TM\phantom{\rule{0.3em}{0ex}}$. By replacing each point in $N$ with the projective space (real or complex) of lines normal to $N\phantom{\rule{0.3em}{0ex}}$, we obtain the blow-up $\stackrel{̂}{M}\phantom{\rule{0.3em}{0ex}}$. Replacing $M$ with $\stackrel{̂}{M}$ amounts to a surgery on the neighborhood of $N$ which alters the topology of the manifold. The diffeomorphism $f$ induces a canonical diffeomorphism $\stackrel{̂}{f}:\stackrel{̂}{M}\to \stackrel{̂}{M}\phantom{\rule{0.3em}{0ex}}$. We prove that under certain assumptions on the local dynamics of $f$ at $N$ the diffeomorphism $\stackrel{̂}{f}$ 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.

##### Keywords
partially hyperbolic diffeomorphism, surgery, blow-up, Anosov submanifold, fiberwise Anosov diffeomorphism
Primary: 37D30