and not to $\bigcup_{j=1}^l U_j$ directly). Now, by using \[it:existsU\]) if $d_j$ is homologically visible, we know that $U_j \cap E^{-1}([0,\kappa_0))$ is connected and consequently we can replace $(\phi_\tau\circ\beta)|_{[t_r,t'_r]}$ by a path in $E^{-1}([0,\kappa_0))$ with the same endpoints. After $m$ steps we obtain a homotopy $\hat\beta:[0,1]\to\Omega$ such that $E(\hat\beta_t)<\kappa_0$ when $\hat\beta_t\in K$. Since $(\phi_\tau\circ\beta)_0,(\phi_\tau\circ\beta)_1
\notin K$ it follows that $(\phi_\tau\circ\beta)_0,(\phi_\tau\circ\beta)_1 \notin
\bigcup_{j=1}^l U_j$ and therefore $\hat\beta_0 \in P_n^-$ ,$\hat\beta_1 \in
P_n^+$, hence $\hat\beta\in\Pi$. This contradicts the minimality of $\kappa_0$.

Contractible and intersecting closed geodesics {#se:contractible}
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Here $M$ still denotes a complete Riemannian cylinder. We assume that there exists a contractible closed geodesic $c\in\Lambda M$. Let us consider the unbounded components of $M\setminus c(S^1)$. Since $c(S^1)$ is bounded, there are at most two distinct unbounded components. If there are two distinct unbounded components $C_-$ and $C_+$, one can assume that $C_-$ is a neighborhood of $-\infty$ and $C_+$ is a neighborhood of $+\infty$. By $C_\pm$ we will mean any of these two neighborhoods. Then $\partial C_\pm$ is a broken geodesic with angles strictly less than $\pi$ inside $C_\pm$ since $c$ is a closed geodesic (see Figure \[fig:label\] for an instance of $\partial C_+$). Hence $C_\pm$ is locally convex. Moreover if 