Limit points of lines of minima in Thurston's boundary of Teichmüller space

Abstract

Given two measured laminations $\mu$ and $\nu$ in a hyperbolic surface which fill up the surface, Kerckhoff [Lines of Minima in Teichmueller space, Duke Math J. 65 (1992) 187–213] defines an associated line of minima along which convex combinations of the length functions of $\mu$ and $\nu$ are minimised. This is a line in Teichmüller space which can be thought as analogous to the geodesic in hyperbolic space determined by two points at infinity. We show that when $\mu$ is uniquely ergodic, this line converges to the projective lamination $\left[\mu \right]$, but that when $\mu$ is rational, the line converges not to $\left[\mu \right]$, but rather to the barycentre of the support of $\mu$. Similar results on the behaviour of Teichmüller geodesics have been proved by Masur [Two boundaries of Teichmueller space, Duke Math. J. 49 (1982) 183–190].