Magnetic flows on Sol-manifolds: dynamical and symplectic aspects

Details Magnetic flows on Sol-manifolds: dynamical and symplectic aspects Magnetic flows on Sol-manifolds: dynamical and symplectic aspects

2007-08-14

arxiv.org/abs/0708.1938v3

Mathematics

Dynamical Systems

Final version to appear in CMP. Two new remarks have been added as well as some numerical calculations for metric entropy

Scientific paper

10.1007/s00220-008-0645-8

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.

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