Magnetic flows on Sol-manifolds: dynamical and symplectic aspects

TL;DR

This paper studies magnetic flows on Sol-manifolds, revealing they have positive entropy and are non-integrable, contrasting with geodesic flows, and demonstrates displaceability of compact sets in twisted cotangent bundles.

Contribution

It provides new insights into the dynamical complexity of magnetic flows on Sol-manifolds and their symplectic properties, highlighting differences from geodesic flows.

Findings

Magnetic flows on Sol-manifolds have positive Liouville entropy.

These flows are never completely integrable.

Compact sets in certain twisted cotangent bundles are displaceable.

Abstract

We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable.