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Magnetic Flows on Sol-Manifolds: Dynamical and Symplectic Aspects

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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.

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Authors and Affiliations

  1. School of Mathematics and Maxwell Institute for Mathematical Sciences, The University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Edinburgh, EH9 3JZ, UK

    Leo T. Butler

  2. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK

    Gabriel P. Paternain

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  1. Leo T. Butler
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  2. Gabriel P. Paternain
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Correspondence to Leo T. Butler.

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Communicated by P. Sarnak

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Butler, L.T., Paternain, G.P. Magnetic Flows on Sol-Manifolds: Dynamical and Symplectic Aspects. Commun. Math. Phys. 284, 187–202 (2008). https://doi.org/10.1007/s00220-008-0645-8

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  • DOI: https://doi.org/10.1007/s00220-008-0645-8

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