in Geometry, Algebra and Dynamics”, funded by the Deutsche Forschungsgemeinschaft.

Preliminaries {#se:preliminaries}
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In this section, we recall some results of Riemannian geometry that we will use in our proofs and fix some notation. For the extension of these notions to the Finsler case, the reader may for instance look at [@CJ13 Section 2].

The energy functional
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Given a complete Riemannian manifold with boundary $M$, we denote by $\Lambda M$ the space of $H^1$-maps $S^1\to M$. In fact, if one wants to avoid analytic questions, we can always reduce our space to a finite-dimensional manifold of broken geodesics. For $\gamma\in\Lambda M$ and $m\in{{\mathbb{N}}}^*$, the iterated loop $\gamma^m\in\Lambda M$ is defined by $t\mapsto \gamma(mt)$. A geodesic is an immersed path $\gamma:{{\mathbb{R}}}\to M$ such that $$\nabla_{\dot{\gamma}}\dot{\gamma} = 0,$$ where $\nabla$ denotes the Levi-Civita connection of the metric and $\dot{\gamma}$ stands for the derivative of $\gamma$. Therefore, in our convention, geodesics have constant speed. A closed geodesic is a geodesic $\gamma$ which is periodic: $\gamma(t+1)=\gamma(t)$ so that $\gamma\in\Lambda M$. Closed geodesics of $M$ are the critical points with non-zero critical value of the energy functional $E:\Lambda M\to [0,+\infty)$, $$E(\gamma) := \int_{S^1} g_{\gamma}(\dot{\gamma},\dot{\gamma}){\mathrm{d}}t,
    \quad \forall \gamma\in\Lambda M.$$ The energy functional $E$ is $C^2$. 