Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

Abstract

Abstract. In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation.

The Conley Conjecture and Beyond

“…Without a distinguished class of short orbits to work with, one is forced to consider all periodic orbits and, already for M = T 2 , use the Conley conjecture type results in place of the Arnold conjecture. Hypothetically, as is observed in Ginzburg and Gürel (2009a), every low energy level should carry infinitely many simple periodic orbits, at least when (M, σ ) is a symplectic CY manifold. This is still a conjecture when dim M > 2, but in dimension two the question has been recently settled in .…”

“…In this setting, in all dimensions, the existence of at least one closed orbit with contractible projection to M was proved in Ginzburg and Gürel (2009a) and Usher (2009).…”

“…This fact does not really require M to be symplectic; it is sufficient to assume that σ = 0. Then a Sturm theory type argument is used in Ginzburg and Gürel (2009a) and Usher (2009) to show that long orbits must necessarily have high index, and hence, by the Arzela-Ascoli theorem, every low energy level carries a periodic orbit. At this step, the assumption that the Hessian d 2 K is positive definite on the normal bundle to M becomes essential, cf.…”

The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

“…However, existence of contractible magnetic geodesics for every low energy levels still holds as Usher proves in [Ush09] building upon previous work of Ginzburg and Gürel [GG09] (see also [Ker99] for multiplicity results, generalizing [Gin87], when σ is a Kähler form).…”

Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level

“…One of the most general results in higher dimension is the existence of one contractible periodic orbit on almost every low energy level if σ = 0 [Sch06]. Existence for every low energy can be obtained by requiring σ to be symplectic [GG09,Ush09]. Finally, multiplicity results hold if σ is a Kähler form [Ker99].…”