Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation

Abstract

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...

Limits of Teichmüller geodesics in the Universal Teichmüller space

“…In the finite-dimensional setting the existence of geodesic Teichmüller rays with limit sets larger than a single point was first demonstrated by Lenzhen in [13]. Such rays and their limit sets in P ML(S) were further investigated by Leininger-Lenzhen-Rafi [12] and Chaika-Masur-Wolf [4].…”

Limit Sets of Weil–Petersson Geodesics

“…Sequences of curves. In [LLR13] and [BLMR16] the authors studied infinite sequences of curves on a surface that limit to non-uniquely ergodic laminations. The novelty in this work is that local estimates on subsurface projections and intersection numbers are promoted to global estimates on these quantities.…”

“…In [Mas82], Masur showed that the Thurston boundary and the Teichmüller visual boundary are not so different, proving that almost every Teichmüller ray converges to a point on the Thurston boundary (though positive dimensional families of rays based at a single point can converge to the same point). Lenzhen [Len08] constructed the first examples of Teichmüller geodesic rays that do not converge to a unique point in the Thurston boundary, and recent constructions have illustrated increasingly exotic behavior [LLR13,BLMR16,CMW14,LMR16].…”

Limit sets of Weil–Petersson geodesics with nonminimal ending laminations

“…A number of authors have studied the limiting behavior of Teichmüller geodesics in relation to the Thurston compactification of Teichmüller space, [Mas82,Ker80] [ Len08,LM10], [LLR13,CMW14], [BLMR16a,LMR16]. This work has highlighted the delicate relationship between the vertical foliation of the quadratic differential defining the geodesic and the limit set in the Thurston boundary.…”