We construct a Teichmüller geodesic which does not have a limit on the Thurston boundary of the Teichmüller space. We also show that for this construction the limit set is contained in a one-dimensional simplex in PMF . 30F60, 32G15; 32F45, 57M50

Abstract. We study the asymptotic geometry of Teichmüller geodesic rays. We show that, when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, the rays diverge in Teichmüller space.

Considering the Teichmüller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are stable. Our main tool is to show that one can get a good estimate for the Lipschitz distance by considering the length ratio of finitely many curves.

We describe a method for constructing Teichmüller geodesics where the vertical foliation ν is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters to ensure that the set of accumulation points of such a geodesic in the Thurston boundary is exactly the projective 1-simplex of all projective measured foliations that are topologically equivalent to ν. With further adjustment of the parameters, one can further assume that the transverse measure is an ergodic measure on the non-uniquely ergodic foliation ν.(see Section 3 for detailed description).Theorem 1.1. There exists R > 0 so that if the powers r i are larger than R, then the path {γ i } is a quasi-geodesic in the curve complex and hence the limitingWe are interested in understanding a Teichmüller geodesic where ν is topologically equivalent to the vertical foliation of the associated quadratic differential. (Recall that there is a one-to-one correspondence between measured laminations and singular measured foliations; see Section 2.7). Let ∆(ν) ⊂ PML(S) be the simplex of all possible projectivized measures on ν. In the case of the five-times punctured sphere, if ν is minimal and not uniquely ergodic then ∆(ν) is one dimensional, that is, it is homeomorphic to an interval. The endpoints of this interval are projective classes associated to ergodic measures on ν, denoted ν α and ν β . Every other measure takes the form ν = c α ν α + c β ν β for positive real numbers c α and c β . Note that the projective class of ν depends only on the ratio of c α and c β . Now fix any point X in Teichmüller space. By a theorem of Hubbard-Masur [HM79], there is a unique Teichmüller geodesicstarting from X so that the vertical foliation associated to g is in the projective class of ν. We examine the limit set Λ(g) of this geodesic in the Thurston boundary of Teichmüller space.By appealing to the results of the third author in [Raf05, Raf14], we can determine how the coefficients {r i } effect the behavior of this Teichmüller geodesic. In particular, at any time t, we can describe the geometry of the surface g(t) using the numbers r i . As a result, we can control the limit set of g. Theorem 1.2. Let ν = c α ν α + c β ν β and g = g(X, ν) be as above. If {r i } satisfies certain growth conditions, (see Definition 7.1 where n i = r 2i−1 and m i = r 2i ) then Λ(g) = ∆(ν), for any value of c α and c β .Note that, in particular, even when ν = ν α is an ergodic measure, the limit set still includes the other ergodic measure ν β . This contrasts with the work of Lenzhen-Masur, where they show that the geodesics g α = g(X, ν α ) and g β = g(X, ν β ) diverge in Teichmüller space, that is the distance in Teichmüller space between g α (t) and g β (t) goes to infinity as t → ∞ [LM10].History and related results. The existence of minimal, filling and non-uniquely ergodic foliations has been known for a long time due to work of Keane [Ke...