Stars at infinity in the Thurston boundary

We study the stars at infinity in the Thurston boundary of Teichmüller space for the Thurston and Teichmüller metrics. For the Thurston metric, we prove that for every finite-type surface, the star of a projective measured lamination is exactly its zero set, extending a theorem of Liu-Shi from closed surfaces. Moreover, the based star agrees with the star. For the Teichmüller metric, we show that both the based star and the star of a projective measured lamination coincide with its two-step zero set. This set can be strictly larger than the zero set, disproving a conjecture of Karlsson.