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The contact property for symplectic magnetic fields on $S^{2}$

Published online by Cambridge University Press:  10 November 2014

GABRIELE BENEDETTI
GABRIELE BENEDETTI*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK email [email protected]
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Abstract

In this paper we give some positive and negative results about the contact property for the energy levels ${\rm\Sigma}_{c}$ of a symplectic magnetic field on $S^{2}$. In the first part we focus on the case of the area form on a surface of revolution. We state a sufficient condition for an energy level to be of contact type and give an example where the contact property fails. If the magnetic curvature is positive, the dynamics and the action of invariant measures can be numerically computed. The collected data hint at the conjecture that an energy level of a symplectic magnetic field with positive magnetic curvature should be of contact type. In the second part we show that, for a small energy $c$, there exist a convex hypersurface $N_{c}$ in $\mathbb{C}^{2}$ and a double cover $N_{c}\rightarrow {\rm\Sigma}_{c}$ such that the pull-back of the characteristic distribution on ${\rm\Sigma}_{c}$ is the standard characteristic distribution on $N_{c}$. As a corollary, we prove that there are either two or infinitely many periodic orbits on ${\rm\Sigma}_{c}$. The second alternative holds if there exists a contractible prime periodic orbit.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 3 , May 2016 , pp. 682 - 713
Copyright
© Cambridge University Press, 2014 

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