On the contact type conjecture for exact magnetic systems
Lina Deschamps, Levin Maier, Tom Stalljohann
TL;DR
The paper resolves the contact type conjecture for a broad class of exact magnetic systems by introducing strong geodesic type and constructing infinite-dimensional families of exact magnetic fields that force null-homologous embedded periodic orbits on all energy levels below the Mañé threshold. It provides explicit formulas for Mañé critical values, showing $c_u(M,g,d\alpha)=c_0(M,g,d\alpha)=\tfrac12\Vert \alpha\Vert_\infty^2$ when a contractible strong geodesic-type loop exists, and proves arbitrarily large multiplicities of such orbits. The results yield that energy surfaces are not of contact type for $\kappa\in(0,c_0]$ in these families, while offering constructive, implementable methods to generate many dynamical behaviors. The authors connect the framework to Lyusternik–Fet theory and the Weinstein conjecture, illustrating rich dynamics on non-aspherical manifolds and on manifolds where Weinstein-type results hold, thereby bridging variational, geometric, and dynamical perspectives in magnetic systems with potential implications for symplectic topology and Hamiltonian dynamics.
Abstract
In this article, we answer-for a class of magnetic systems-a question now known as the contact type conjecture, whose origin trace back to the 1998 work of Contreras, Iturriaga, Paternain, and Paternain. For a broad class of magnetic systems, we explicitly construct, on any closed manifold, an infinite-dimensional space of exact magnetic systems, which we refer to as magnetic systems of strong geodesic type. For each such system, there exists at least one null-homologous embedded periodic orbit on every energy level, with negative action for energies below the strict Mañé critical value. As a consequence, the corresponding energy surfaces are not of contact type below this threshold. Thus, for this class of systems, the contact type conjecture holds true. Moreover, for these systems, both the strict and the lowest Mañé critical values can be computed explicitly, and they coincide whenever the aforementioned periodic magnetic geodesic is contractible, without requiring any additional assumptions on the manifold. Several remarkable multiplicity results also hold, guaranteeing arbitrarily large numbers of embedded null-homologous periodic magnetic geodesics on every energy level. We illustrate the richness of this class through two types of examples. First, on any non-aspherical manifold, there exists a dense subset of the space of Riemannian metrics such that, for each such metric, one can construct an infinite-dimensional space of exact magnetic fields yielding magnetic systems of strong geodesic type. Second, on any closed contact manifold for which the strong Weinstein conjecture holds, one can construct an infinite-dimensional space of Riemannian metrics such that, for each such metric, the magnetic system induced by the fixed contact form is of strong geodesic type.