thus get a family of closed geodesics which are not homotopic and not iterations of each other.

Now that Theorem \[thm:main\] is proved under hypothesis \[it:contractible\], in order to prove it when there is one self-intersecting closed geodesic $c$ or two intersecting ones $c_1$ and $c_2$, one can assume that these geodesics are not contractible. Therefore, in both respective cases, $M\setminus c(S^1)$ or $M\setminus(c_1(S^1)\cup c_2(S^1))$ has exactly two unbounded connected components $C_-$ and $C_+$, which are locally convex by construction. The intersection hypothesis then implies that none of the boundaries $\partial C_\pm$ is totally geodesic. Hence the conclusion follows by applying Theorem \[thm:Bangert\].

Geodesic of non-zero average index {#se:mind}
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We assume that there exists a closed geodesic $c\in\Lambda M$ of average index $\operatorname{\overline{ind}}(c) > 0$. If $c$ is contractible or self-intersecting, we already know that there are infinitely many closed geodesics. Let us assume that $c$ is an embedded curve generating $\pi_1(M)\simeq{{\mathbb{Z}}}$. By a slight abuse of notation, we identify the loop $c:S^1\to M$ with its lift ${{\mathbb{R}}}\to M$.

\[lem:delta\] There exist $k\in{{\mathbb{N}}}^*$ and $\delta \in (0,1]$ such that for all $s\in {{\mathbb{R}}}$, there exists a Jacobi field $J:{{\mathbb{R}}}\to c^*TM$ of $c$ with

1.  \[it:Js\] $J(s)=0$,

2.  $J|_{(s,s+\delta)}$ non-vanishing,

