Contact Type Conjecture in Magnetic Systems

• Contact Type Conjecture is a fundamental concept in symplectic and contact topology, asserting that energy hypersurfaces in certain magnetic Hamiltonian systems cannot be of contact type.
• The conjecture hinges on the existence of null-homologous periodic orbits with negative action, which create invariant measures violating conditions for contact type energy surfaces.
• The local construction and multiplicity of periodic orbits in exact magnetic systems provide robust counterexamples, refining our understanding of critical Mañé values and global dynamics.

The Contact Type Conjecture is a central question in symplectic and contact topology concerning the classification of energy hypersurfaces with contact type in Hamiltonian systems, especially those arising in magnetic and twisted geodesic flows. In its most impactful formulation, it asserts that under certain geometric and dynamical conditions, specific energy levels—particularly those below a critical value in certain exact magnetic systems—cannot be of contact type. This property is deeply tied to the existence of invariant measures supported on orbits with negative action and thus has significant implications for the theory of periodic orbits, symplectic rigidity, and the global dynamics of Hamiltonian systems.

1. Construction of Magnetic Systems of Strong Geodesic Type

For a closed manifold $M$ of dimension $m \geq 3$, an exact magnetic system is defined as a triple $(M, g, d\alpha)$, where $g$ is a Riemannian metric and $\alpha$ is a 1-form such that the magnetic field $\sigma = d\alpha$ is exact. The framework of magnetic systems of strong geodesic type is synthesized as follows (Deschamps et al., 1 Aug 2025):

• One prescribes a smooth closed curve $\gamma \subset M$ and constructs a tubular neighborhood $U_\gamma \cong S^1 \times \mathbb{R}^{m-1}$ around $\gamma$.
• The 1-form $\alpha$ is chosen locally so that its $g$-metric dual vector field $V^\alpha$ matches the tangent direction of $\gamma$: for all $t\in S^1$, in the coordinates of $U_\gamma$,

$V^\alpha(t,0) = G(t,0)\cdot e_1,$

where $G$ is the $g$-metric matrix.

• The norm $|\alpha_{\gamma(t)}|_g = \|\alpha\|_\infty$ is constant along $\gamma$, and the duality $$g_{\gamma(t)}(\dot\gamma(t),v) = \alpha_{\gamma(t)}(v)$$ holds for all $v\in T_{\gamma(t)}M$.
• Away from $U_\gamma$ and outside a compact set, both $g$ and $\alpha$ can be freely perturbed, producing infinite-dimensional families of such systems.

This local construction is adaptable to arbitrary closed manifolds and is especially robust since it can be iterated on any finite collection of disjoint, null-homologous loops.

2. Existence of Null-Homologous Periodic Orbits and Negative Action

For each exact magnetic system of strong geodesic type, a null-homologous embedded periodic orbit $\gamma$ exists on every energy level $\Sigma_\kappa$ (Deschamps et al., 1 Aug 2025). Key properties include:

• Negative action: For $\kappa$ below a critical threshold, $\gamma$ minimizes the action

$$S_{L+\kappa}(\gamma) = \int_0^T \left[ \frac{1}{2}|\dot\gamma(t)|_g^2 - \alpha_{\gamma(t)}(\dot\gamma(t)) + \kappa \right] dt < 0,$$

where $L(x, v) = \frac{1}{2}|v|_g^2 - \alpha_x(v)$.

• The support of $\gamma$ gives rise to an invariant measure $\mu$ satisfying $\int_{\Sigma_\kappa} \lambda(Z_E) d\mu = 0$, where $\lambda$ is the Liouville 1-form and $Z_E$ is the Hamiltonian vector field.

These properties leverage the criterion for contact type energy surfaces: an energy surface $\Sigma_\kappa$ is of contact type if every invariant probability measure has $\int_{\Sigma_\kappa} \lambda(Z_E) d\mu \neq 0$. If there exists any invariant measure with vanishing average, $\Sigma_\kappa$ fails to be of contact type.

3. Contact Type Property of Energy Surfaces and Critical Values

A hypersurface $\Sigma_\kappa$ fails to be of contact type on energy levels $\kappa \in (0, c_0]$, where $c_0$ is the strict Mañé critical value (Deschamps et al., 1 Aug 2025). For magnetic Lagrangians,

$$c_0(L) = \inf \left\{ \kappa \in \mathbb{R} \mid S_{L+\kappa}(\gamma) \geq 0\, \forall\; \text{null-homologous closed curves}~\gamma \right\}.$$

The case of contractible geodesics is particularly sharp:

• If the constructed $\gamma$ is contractible, then the lowest Mañé critical value $c_u$ equals $c_0$: $c_u = c_0$.
• Explicit computation yields $c_0(M, g, \alpha) = \frac{1}{2}\|\alpha\|_\infty^2$.

This explicit calculation of critical values under contractibility represents an advance, removing previous topological obstructions or nondegeneracy hypotheses.

4. Multiplicity of Periodic Orbits

One of the major results is the multiplicity phenomenon: for any finite $n$, one can construct, by iterating the local argument on $n$ pairwise disjoint, coorientable, null-homologous embedded loops, an exact magnetic system of strong geodesic type with at least $n$ such null-homologous embedded periodic orbits on each energy level. Each orbit supports an invariant measure providing a certificate of non-contact type for $\Sigma_\kappa$ for $\kappa \leq c_0$ (Deschamps et al., 1 Aug 2025).

The construction is flexible and yields an infinite-dimensional family of such magnetic systems, with arbitrary prescribed multiplicity of periodic orbits.

5. Model Examples

Table: Settings for Strong Geodesic Type Magnetic Systems

Setting	Construction Description	Implication
Non-aspherical closed manifold	Dense subset of metrics yields infinite-dimensional space of $d\alpha$	Each energy level: periodic orbit
Closed contact manifold, strong Weinstein holds	Infinite-dimensional set of Riemannian metrics, fixed contact form induces magnetic system	Rich Reeb dynamics yields periodic orbits

In the non-aspherical case, a dense subset of Riemannian metrics exists for which one obtains infinite-dimensional families of strong geodesic type systems. On closed contact manifolds satisfying the strong Weinstein conjecture, the rich structure of the Reeb flow is transferred to the magnetic system.

6. Broader Implications and Applications

These results establish the Contact Type Conjecture in full generality for exact magnetic systems of strong geodesic type: the energy hypersurfaces below the critical threshold are never of contact type (Deschamps et al., 1 Aug 2025). The consequences are significant:

• The phenomena apply in dimensions $m\geq3$ and for a broad class of closed manifolds, showing that the obstructions to contact type are intrinsic to the dynamical system—rooted in the existence of periodic orbits with negative action.
• The explicit invariant measure argument, and the ability to prescribe local geometry, provide a toolkit for constructing further explicit counterexamples to contact type in Hamiltonian systems.
• The coincidence of the strict and lowest Mañé critical values in contractible cases helps delineate a sharp boundary in parameter space for contact type, with implications for symplectic and contact geometry, dynamical systems, and variational calculus.
• The multiplicity result not only gives a single "bad" energy level but in fact demonstrates arbitrary multiplicity, strengthening non-contact type results via the abundance of periodic orbits with negative action.

The construction methods and arguments are highly local and flexible, suggesting applicability to wider families of Hamiltonian systems, including those arising in high-dimensional mechanics, magnetic flows, and geometric control theory.

7. Outlook

The explicit and infinite-dimensional constructions given for exact magnetic systems and the resulting theorems confirm the Contact Type Conjecture for a broad range of systems, making the existence of non-contact type energy levels a generic and robust feature in these settings. The approaches developed may further illuminate additional rigidity versus flexibility dichotomies in symplectic and contact topology and should provide templates for similar results in related classes of Hamiltonian and Lagrangian systems.