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

Gogolev, Andrey¹

¹ 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

Geometry & topology, Tome 22 (2018) no. 4, pp. 2219-2252.

Résumé

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

Détail

DOI : 10.2140/gt.2018.22.2219

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

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