The contact property for magnetic flows on surfaces
This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the attention to low energy levels on S^2 with a symplectic σ and we show that these levels are of dynamically convex contact type. Hence, we prove that, in the non-degenerate case, there exists a Poincare section of disc-type and at least an elliptic periodic orbit. In the general case, we show that there are either 2 or infinitely many periodic orbits on Σ_m and that we can divide the periodic orbits in two distinguished classes, short and long, depending on their period. Then, we look at the case of surfaces of revolution, where we give a sufficient condition for the existence of infinitely many periodic orbits. Finally, we discuss a generalisation of dynamical convexity introduced recently by Abreu and Macarini, which applies also to surfaces with genus at least two.
Magnetic flows, Symplectic geometry, Periodic orbits
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