The asymptotic behavior of Teichm\"uller rays

Abstract

In this paper, we consider the asymptotic behavior of two Teichm\"uller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem in \cite{Amano14}. We also consider the infimum of the asymptotic distance in shifting base points of the rays along the geodesics. We show that the infimum is represented by two quantities. One is the detour metric between the end points of the rays on the Gardiner-Masur boundary of the Teichm\"uller space, and the other is the Teichm\"uller distance between the end points of the rays on the augmented Teichm\"uller space.