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

Abstract

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 $\tilde{\mathcal{L}}$.

arXiv comment20 pages without appendix, 26 pages with appendix, 9 figures. Superseded by arXiv:2210.03937. This posting is kept intact since it contains some additional constructions omitted from arXiv:2210.03937