### Abstract

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.