Stable weakly shadowable volume-preserving systems are volume-hyperbolic

TL;DR

This paper proves that stable weakly shadowable volume-preserving systems on compact manifolds exhibit volume-hyperbolic dominated splittings, leading to global hyperbolicity in low dimensions, extending to divergence-free vector fields.

Contribution

It establishes that C1-stably weakly shadowable volume-preserving systems necessarily have volume-hyperbolic dominated splittings, a significant structural insight.

Findings

Systems have a dominated splitting E + F

Both E and F are volume-hyperbolic

In low dimensions, systems are globally hyperbolic

Abstract

We prove that any C1-stably weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E + F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.