In this paper, we obtain the explicit limit value of the Teichmüller distance between two Teichmüller geodesic rays which are determined by Jenkins-Strebel differentials having a common end point on the augmented Teichmüller space. Furthermore, we also obtain a condition under which these two rays are asymptotic. This is similar to a result of Farb and Masur.2000 Mathematics Subject Classification. Primary 32G15, Secondary 30F60.

Abstract. In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem by the author in On behavior of pairs of Teichmüller geodesic rays, 2014 . We also consider the infimum of the asymptotic distance up to choice of 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 Teichmüller space, and the other is the Teichmüller distance between the end points of the rays on the augmented Teichmüller space.

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In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem in [Ama14]. 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 Teichmüller space, and the other is the Teichmüller distance between the end points of the rays on the augmented Teichmüller space. IntroductionLet X be a Riemann surface of genus g with n punctures such that 3g −3+n > 0, and T (X) be the Teichmüller space of X. Any Teichmüller geodesic ray on T (X) is determined by a holomorphic quadratic differential on an starting point of the ray. A geodesic ray is called a Jenkins-Strebel ray if it is given by a Jenkins-Strebel differential. In [Ama14], we obtain a condition for two Jenkins-Strebel rays to be asymptotic (Corollary 1.2 in [Ama14]). To obtain this condition, we use Theorem 1.1 in [Ama14] which gives the explicit asymptotic value of the Teichmüller distance between two similar Jenkins-Strebel rays with the same end point in the augmented Teichmüller space. In this paper, we improve this theorem, and obtain the asymptotic value of the distance between any two Jenkins-Strebel rays.Let r, r ′ be Jenkins-Strebel rays on T (X) from r(0) = [Y, f ], r(0) ′ = [Y ′ , f ′ ] determined by Jenkins-Strebel differentials q, q ′ with unit norm on Y , Y ′ respectively. It is known (cf.[HS07]) that the Jenkins-Strebel rays r, r ′ have limits, say r(∞), r ′ (∞), on the boundary of the augmented Teichmüller space T (X). Suppose that r, r ′ are similar, that is, there exist mutually disjoint simple closed curves γ 1 , • • • , γ k on X such that the set of homotopy classes of core curves of the annuli corresponding to q, q ′ are represented by f (γ 1 ),on Y ′ respectively. We denote by m j , m ′ j the moduli of the annuli on Y , Y ′ with core curves homotopic to f (γ j ), f ′ (γ j ) respectively. We can define the Teichmüller distance d T (X) (r(∞), r ′ (∞)) between the end points r(∞), r ′ (∞).