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

Let $\mathcal{L}$ be a measured geodesic lamination on a complete hyperbolic surface of finite area. Assuming $\mathcal{L}$ is not a multicurve, our main result establishes the existence of a geodesic ray which has finite intersection number with $\mathcal{L}$ but is not asymptotic to any leaf of $\mathcal{L}$ nor eventually disjoint from $\mathcal{L}$. In fact, we show that the 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 the halo of $\tilde{\mathcal{L}}$), which is disjoint from the endpoints of leaves of the lifted lamination $\tilde{\mathcal{L}}$.