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

TL;DR

This paper proves the existence of special geodesic rays on hyperbolic surfaces that intersect a measured lamination finitely but are not asymptotic to its leaves, revealing a complex boundary structure called the halo.

Contribution

It establishes the existence of geodesic rays with finite intersection to a lamination that are neither asymptotic nor disjoint, and characterizes their endpoints forming an uncountable halo.

Findings

Existence of such geodesic rays with finite intersection

The halo of the lamination is uncountable and disjoint from leaf endpoints

The halo provides new insights into the boundary behavior of laminations

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