E^{-1}([0,E(\gamma))$ is not connected. For the proof of Theorem \[thm:Bangert\], we need the following statement, which tells us that isolated closed geodesics cannot remain mountain pass critical points of the energy functional when sufficiently iterated. A geometric proof is given by Bangert [@Ban80].

\[thm:mountainpass\] Let $\gamma$ be an isolated closed geodesic on $M$, where $\dim M=2$. Then there exists an integer $m_\gamma\in{{\mathbb{N}}}$ such that the following holds: For all integer $m\in{{\mathbb{N}}}$ with $m\ge m_\gamma$, there is a neighborhood $U$ of $S^1
    \cdot \gamma$ in $\Lambda M$ such that $U \cap E^{-1}([0,E(\gamma^m)))$ is connected.

According to Gromoll-Meyer [@GM68], given an isolated closed geodesic $\gamma$, there exists a connected neighborhood $U\subset\Lambda M$ of the critical circle $S^1\cdot\gamma$ such that $${\textup{C}}_*(S^1\cdot\gamma)\simeq H_*\left(U,U\cap E^{-1}([0,E(\gamma))\right).$$ If $\gamma$ and all its iterates are homologically invisible, Theorem \[thm:mountainpass\] is thus true for $m_\gamma = 1$.

Assume there are only finitely many prime closed geodesics $\gamma_1,\ldots,\gamma_k$ which have homologically visible iterates and which intersect $M\setminus (C_- \cup C_+)$. We will now derive a contradiction from this assumption. We will define a suitable finite-dimensional approximation $\Omega=\Omega(\mathcal{O},\kappa,j)$. Now as the statement of Theorem \[thm:mountainpass\] remains true in a finite-dimensional approximation, we get that there exists $m_0\in{{\mathbb{N}}}$ such that for all 