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

TL;DR

This paper introduces a method to generate new partially hyperbolic diffeomorphisms by performing blow-ups along invariant submanifolds, extending the class of known examples through topological surgeries.

Contribution

It develops a systematic approach to construct new partially hyperbolic systems via blow-ups and connected sums, expanding the toolkit for studying such dynamical systems.

Findings

Blow-up construction preserves partial hyperbolicity under certain conditions.

Connected sum techniques enable combining different partially hyperbolic systems.

Several examples demonstrate the applicability of the methods.

Abstract

We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism $f\colon M\to M$ 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 $\hat M$. 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 $\hat f\colon \hat M\to \hat M$. 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.