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.
Similar content being viewed by others
References
Bolsinov A.V., Dullin H.R., Veselov A.P.: Spectra of Sol-manifolds: arithmetic and quantum monod- romy. Commun. Math. Phys. 264, 583–611 (2006)
Bolsinov A.V., Taimanov I.A.: Integrable geodesic flows with positive topological entropy. Invent. Math. 140, 639–650 (2000)
Burns K., Paternain G.P.: Anosov magnetic flows, critical values and topological entropy. Nonlinearity 15, 281–314 (2002)
Cieliebak K., Frauenfelder U., Paternain G.P.: Symplectic topology of Mañé’s critical value. In preparation
Ginzburg, V.L., Kerman, E.: Periodic orbits in magnetic fields in dimensions greater than two. Geometry and Topology in Dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemp. Math. 246. Providence, RI: Amer. Math. Soc., 1999, pp. 113–121
Gromov M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)
Hattori A.: Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I 8, 289–331 (1960)
Laudenbach F., Sikorav J.-C.: Hamiltonian disjunction and limits of Lagrangian submanifolds. Internat. Math. Res. Notices 4, 161–168 (1994)
Siegel C.L., Moser J.: Lectures on Celestial Mechanics. Springer-Verlag, New York (1971)
Nomizu K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math. 59, 531–538 (1954)
Paternain G.P.: Magnetic rigidity of horocycle flows. Pacific J. Math. 225, 301–323 (2006)
Polterovich, L.: An obstacle to non-Lagrangian intersection. In: The Floer memorial volume, Progr. Math. 133. Basel: Birkhäuser, 1995, pp. 575–586
Schlenk F.: Applications of Hofer’s geometry to Hamiltonian dynamics. Comment. Math. Helv. 81, 105–121 (2006)
Scott P.: The Geometries of 3-manifolds. Bull. London Math. Soc. 15, 401–487 (1983)
Wall C.T.C.: Geometric structures on compact complex analytic surfaces. Topology 25, 119–153 (1986)
Zung N.T.: Convergence versus integrability in Birkhoff normal form. Ann. Math. 161, 141–156 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Sarnak
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0645-8