said to be homologically invisible otherwise. We will be interested in the properties of the local homology ${\textup{C}}_*(S^1\cdot \gamma)$ only in the case where $\gamma$ is a closed geodesic of average index $\operatorname{\overline{ind}}(\gamma)=0$ and whose image $\gamma(S^1)$ lies in the interior of $M$ ($\operatorname{\overline{ind}}(\gamma)=0$ is equivalent to the fact that $\operatorname{ind}(\gamma^m)$ vanishes for all $m\geq 1$). Let $\gamma\in\Lambda M$ be such a closed geodesic. Given $m\in{{\mathbb{N}}}$, we denote by $\psi_m : \Lambda M\to\Lambda M$ the iteration map $\psi_m(\delta):=\delta^m$. According to a theorem of Gromoll-Meyer [@GM69a Theorem 3], the local homology of ${\textup{C}}_d(S^1\cdot \gamma)$ is zero in degrees $d\geq 2\dim M$ and there exists infinitely many positive integers $m$ such that the induced map in homology $$\label{eq:GM}
    (\psi_m)_* : {\textup{C}}_*(S^1\cdot \gamma) \to {\textup{C}}_*(S^1\cdot \gamma^m)$$ is an isomorphism. On the other hand, a theorem of Bangert-Klingenberg [@BK83 Corollary 1] states that there exists $m_0\in{{\mathbb{N}}}$ above which for all $m\geq m_0$, there exists $e_m > m^2 E(\gamma)$ such that the composition $$\label{eq:BK}
    {\textup{C}}_*(S^1\cdot \gamma) \xrightarrow{(\psi_m)_*}
    {\textup{C}}_*(S^1\cdot \gamma^m) \xrightarrow{\textup{inc}_*}
    H_*\left(\big\{ E < e_m\big\},
    \left\{ E< m^2 E(\gamma) \right\} \right)$$ is zero.

Proof of Bangert theorem {#se:Bangert}
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A closed geodesic $\gamma$ is a mountain pass if, for all neighborhoods $U\subset\Lambda M$ of $S^1\cdot\gamma$, $U\cap 