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Let $\mathcal{L}$ be a measured geodesic lamination on a complete hyperbolicsurface of finite area. Assuming $\mathcal{L}$ is not a multicurve, our mainresult establishes the existence of a geodesic ray which has finiteintersection number with $\mathcal{L}$ but is not asymptotic to any leaf of$\mathcal{L}$ nor eventually disjoint from $\mathcal{L}$. In fact, we show thatthe endpoints of such rays, when lifted to the universal cover $\mathbb{H}^2$of $X$, give an uncountable set $h\tilde{\mathcal{L}}\subset S^1$ (called thehalo of $\tilde{\mathcal{L}}$), which is disjoint from the endpoints of leavesof the lifted lamination $\tilde{\mathcal{L}}$.

Train tracks, entropy, and the halo of a measured lamination