Subharmonic solutions of Hamiltonian equations on tori


Let the torus $ T^{2n}$ be equipped with the standard symplectic structure and a periodic Hamiltonian $\mathcal{H} \in C^{3}(S^{1}\times T^{2n}, \mathbb{R})$. We look for periodic orbits of the Hamiltonian flow $ \dot{\boldsymbol{u}}(t)=J\nabla\mathcal{H} (t,\boldsymbol(t)). $ A subharmonic solution is a periodic orbit with minimal period an integral multiple $ m $ of the period of $\mathcal{H} $, with $ m>1 $.

We prove that if the Hamiltonian flow has only finitely many orbits with the same period as $\mathcal{H}$, then there are subharmonic solutions with arbitrarily high minimal period. Thus there are always infinitely many distinct periodic orbits.

The results proved here were proved in the nondegenerate case by Conley and Zehnder and in the case $ n=1 $ by Le Calvez.