Null-Homologous Embedded Periodic Orbits

• Null-homologous embedded periodic orbits are closed trajectories in magnetic flows whose homology class vanishes, offering a clear framework to study energy hypersurface geometry.
• They are explicitly constructed via reparametrization of magnetic geodesics to achieve any desired energy level, often yielding trajectories with negative action.
• Multiplicity results demonstrate their robustness and interconnection with symplectic topology, variational methods, and the geometry of closed manifolds.

A null-homologous embedded periodic orbit is an embedded closed trajectory in a dynamical or geometric system whose homology class vanishes in the ambient manifold, typically satisfying additional geometric or variational properties dictated by the context (e.g., magnetic geodesics, Reeb flows, Hamiltonian flows). In exact magnetic flows on closed manifolds, recent work establishes the existence and multiplicity of such orbits on all energy levels for a large class of systems, yielding new insights into the geometry of energy hypersurfaces, the structure of the Mañé critical value, and the contact type conjecture (Deschamps et al., 1 Aug 2025).

1. Rigorous Definition and Setting

Given a closed smooth manifold $M$, a Riemannian metric $g$, and a closed 2-form (the "magnetic field") $d\alpha$, an exact magnetic system is the triple $(M, g, d\alpha)$ with primitive 1-form $\alpha$. The magnetic flow corresponds to the dynamics of magnetic geodesics, i.e., solutions $\gamma : S^1 \to M$ of

$$\frac{D\dot{\gamma}}{dt} = Y(\gamma, \dot{\gamma}), \qquad g_{\gamma} (Y(\gamma, v), w) = d\alpha_{\gamma}(v, w).$$

A periodic orbit is null-homologous if its class $[\gamma]=0$ in $H_1(M; \mathbb{R})$, and embedded if its image is an embedded circle.

For "magnetic systems of strong geodesic type," the construction further requires:

• $\gamma$ is a smooth embedded closed curve,
• $\gamma$ is a geodesic for $(M,g)$,
• $d\alpha(\dot\gamma(t), \cdot) = 0$ along $\gamma$,
• $\alpha$ attains its global maximum in $g$-norm along $\gamma$ and satisfies $g_{\gamma}(\dot\gamma, v) = \alpha_{\gamma}(v)$ for $v \in T_{\gamma}M$.

2. Existence on All Energy Levels

For every exact magnetic system of strong geodesic type, there exists at least one null-homologous embedded periodic orbit on every energy level $\kappa>0$ (Deschamps et al., 1 Aug 2025). The construction is explicit:

• Given $\gamma$ as above, define the reparametrized curve $\gamma_r(t) = \gamma(rt)$ for $0
• For any prescribed kinetic energy level $\kappa$, one selects $r$ such that the reparametrized curve satisfies $E(\gamma_r, \dot\gamma_r) = \kappa$ with $E(x,v) = \tfrac{1}{2}g_x(v,v)$.

Because the original curve $\gamma$ is null-homologous, every such reparametrized orbit $\gamma_r$ is also null-homologous on the corresponding energy hypersurface.

3. Action Functional and Contact Type

The action functional for these systems is given by

$$S_{L+\kappa}(\gamma) = \int_0^T \left( \tfrac{1}{2}|\dot{\gamma}(t)|^2 - \alpha_{\gamma(t)}(\dot{\gamma}(t)) + \kappa \right) dt.$$

A fundamental result is that, for energy levels $\kappa$ below the strict Mañé critical value $c_0 = \tfrac{1}{2}\|\alpha\|_\infty^2$, the constructed null-homologous embedded periodic orbit has negative action: $S_{L+\kappa}(\gamma_r) = r(r-1)\|\alpha\|_\infty^2 < 0 \qquad \text{for %%%%30%%%%.}$ This property is critical for the global symplectic geometry of the magnetic flow: according to criteria by Contreras, Macarini, and Paternain, the existence of a null-homologous periodic orbit with negative action at energy level $\kappa$ implies that the corresponding energy hypersurface $\Sigma_\kappa$ is not of contact type.

4. Multiplicity and Richness

Multiplicity results are achieved by constructing families of disjoint, coorientable, null-homologous embedded curves and encoding each as a magnetic geodesic of strong geodesic type. By considering appropriate infinite-dimensional families of metrics and fields, one can ensure that for every energy level $\kappa$ there are arbitrarily large numbers of null-homologous embedded periodic magnetic geodesics with non-positive (or negative) action. These families are robust in the sense that:

• On any closed non-aspherical manifold (i.e., $\pi_k(M) \neq 0$ for $k \geq 2$), a dense set of Riemannian metrics admits this construction.
• On any closed contact manifold where the strong Weinstein conjecture holds, similar multiplicity results are available for an infinite-dimensional family of Riemannian metrics.

Contractibility of the periodic magnetic geodesic ensures that strict and lowest Mañé critical values coincide, and that the negativity of action is robust without extra manifold assumptions.

5. Explicit Computation of Critical Values

In the strong geodesic type case, the following holds:

• The strict Mañé critical value is computed by $c_0(M,g,\alpha) = \tfrac{1}{2}\|\alpha\|_\infty^2$.
• For contractible periodic orbits $\gamma$, the critical value coincides with the lowest Mañé critical value without further assumptions on $M$.

Thus, criteria for the action and properties of the energy level reduce to explicit analytical calculations involving the primitive $\alpha$.

6. Broader Examples and Applications

The construction encompasses numerous situations:

• On non-aspherical closed manifolds, Lyusternik–Fet theory guarantees contractible closed geodesics; these generate (via the above construction) null-homologous embedded periodic orbits on all energy levels for dense sets of metrics and infinite-dimensional families of exact magnetic fields.
• For closed contact manifolds where the strong Weinstein conjecture holds, one can always construct infinitely many such orbits for a suitable metric and exact magnetic field, with the field derived from the contact form.

This demonstrates an abundance of null-homologous embedded periodic orbits in diverse global settings.

7. Significance for Symplectic and Contact Topology

The existence and negativity of action for these orbits resolves the contact type conjecture for broad classes of exact magnetic flows: energy hypersurfaces below the critical value are not of contact type precisely because of the presence of such orbits. The explicit construction of families of these orbits and the calculation of relevant critical values establish strong links between the dynamical, variational, and topological characteristics of the flow and feed directly into the paper of global properties such as the failure of the energy surface to support a contact form.

This framework clarifies both the ubiquity and the geometric role of null-homologous embedded periodic orbits in the context of exact magnetic systems, tying together topology, variational methods, and symplectic geometry (Deschamps et al., 1 Aug 2025).