Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

Brock, Jeffrey; Leininger, Christopher J.; Modami, Babak; Rafi, Kasra

doi:10.48550/arxiv.1601.03368

Cited by 4 publications

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“…In Section 2.5 we state our technical results about sequences of curves on surfaces that limit to non-uniquely ergodic laminations. These results are minor variations of those in [BLMR16], and their proofs are sketched in the appendix of the paper. In Section 3 we construct explicit examples of non-uniquely ergodic laminations on punctured spheres, appealing to the results from Section 2.5.…”

Section: Introductionmentioning

confidence: 61%

“…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.…”

Section: Curves and Laminationsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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Limit Sets of Weil–Petersson Geodesics

Brock

Leininger

Modami

et al. 2018

International Mathematics Research Notices

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Section: Introductionmentioning

confidence: 61%

Section: Curves and Laminationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Limit Sets of Weil–Petersson Geodesics

Brock

Leininger

Modami

et al. 2018

International Mathematics Research Notices

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“…Lenzhen [12] provided the first example of a geodesic ray whose limit set in PMF is more than one point. The study of limit sets in PMF continued in [3] and [11], where the influence of the topological and dynamical properties of the associated vertical foliation is studied, and in [2] and [13], where rays with limits sets homeomorphic to a circle and 2-simplex are constructed, respectively. It would be interesting to know whether the kind of behavior we produce in Theorem 1.1 can occur in PMF.…”

Section: Introductionmentioning

confidence: 99%

Exotic limit sets of Teichmüller geodesics in the HHS boundary

Mousley

2019

Groups Geom. Dyn.

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We answer a question of Durham, Hagen, and Sisto, proving that a Teichmüller geodesic ray does not necessarily converge to a unique point in the hierarchically hyperbolic space boundary of Teichmüller space. In fact, we prove that the limit set can be almost anything allowed by the topology. MSC 2010 Subject Classification: 30F60, 32Q05 (primary), 57M50 (secondary) IntroductionLet S = S g be a connected, closed, orientable surface of genus g ≥ 2, and let T (S) denote the Teichmüller space of S equipped with the Teichmüller metric. Masur [15] proved that T (S) is not non-positively curved in the sense of Busemann, and Masur and Wolf [20] showed that T (S) is not (Gromov) hyperbolic. In this paper, we explore to what extent T (S) has features of negative curvature by studying the asymptotic behavior of geodesics.The notion of Gromov boundary can be generalized in several ways to obtain a boundary for T (S). For example the visual boundary and the Morse boundary agree with the Gromov boundary when the space is hyperbolic, and these boundaries are well-defined for T (S) (see [21] and [5]). Here we consider another generalization. Both hyperbolic spaces and T (S) can be equipped with a geometric structure defined by Behrstock, Hagen, and Sisto [1] called a hierarchically hyperbolic space (HHS) structure. (That T (S) can be equipped with such a structure follows from the results in [7], [8], [19], [27].) These structures were used by Durham, Hagen, and Sisto [6] to construct a boundary, which we will call the HHS boundary (see Section 2 for definitions).Working in the HHS paradigm, the question becomes how do the asymptotics of geodesic rays in the HHS boundary of T (S) compare to those of geodesic rays in the HHS boundary of a hyperbolic space? The identity map on a hyperbolic space extends to a homeomorphism between its HHS and Gromov boundaries, so certainly in this case geodesic rays are wellbehaved. In [6] Durham, Hagen, and Sisto asked for a description of limit sets of Teichmüller geodesic rays in the HHS boundary. Our main result provides an answer to this question.Theorem 1.1. Given a continuous map γ : R → 2 to the standard 2-simplex, there exists a Teichmüller geodesic ray G in T (S 3 ) and an embedding of 2 into the HHS boundary of T (S 3 ) such that the limit set of G in the HHS boundary is the image of γ(R).The study of limiting behaviors of Teichmüller geodesic rays began with Kerckhoff [10]. He proved that the Teichmüller boundary of T (S) (the collection of all geodesic rays emanating from a fixed basepoint) is basepoint dependent. Since then, the limit sets of geodesic rays in Thurston's compactification of T (S) by PMF(S), the space of projectivized measured foliations, have received much attention. Masur [16] showed that almost all Teichmüller geodesic rays converge to a unique point in PMF. Lenzhen [12] provided the first example of a geodesic ray whose limit set in PMF is more than one 1 arXiv:1704.08645v1 [math.GT]

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“…In [LLR13], it is shown that the same phenomenon can take place for a minimal foliation, with limit set being the entire one-dimensional simplex. In [CMW14] an example of minimal foliation is constructed where the limit set of the corresponding ray is a proper subset of a onedimensional simplex of measures and in [BLMR16a] an example is constructed where the limit set is not simply connected and is homeomorphic to a circle. Similar phenomena is also possible for the geodesic in Teichmüller space equipped with the Weil-Petersson metric [BLMR16b,BLMR17].…”

Section: Introductionmentioning

confidence: 99%

Teichmueller geodesics with d-dimensional limit sets

Lenzhen,

Modami,

Rafi

2016

Preprint

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Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

Cited by 4 publications

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Limit Sets of Weil–Petersson Geodesics

Limit Sets of Weil–Petersson Geodesics

Exotic limit sets of Teichmüller geodesics in the HHS boundary

Teichmueller geodesics with d-dimensional limit sets

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