On the contact type conjecture for exact magnetic systems

TL;DR

This paper proves the contact type conjecture for a broad class of magnetic systems by constructing infinite-dimensional spaces of such systems with specific dynamical properties on closed manifolds.

Contribution

It explicitly constructs magnetic systems of strong geodesic type, verifies the conjecture for these systems, and computes critical values explicitly, with multiple multiplicity results.

Findings

Existence of null-homologous periodic orbits on all energy levels below a critical value.

Energy surfaces are not of contact type below the Mañé critical value for these systems.

Infinite-dimensional spaces of magnetic systems of strong geodesic type are constructed on various manifolds.

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 Ma\~n\'e 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 Ma\~n\'e 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.