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

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