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

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 $N\subset M$. 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.