Limit sets of Teichm\"uller geodesics with minimal non-uniquely ergodic vertical foliation
@article{Leininger2013LimitSO, title={Limit sets of Teichm\"uller geodesics with minimal non-uniquely ergodic vertical foliation}, author={Christopher J. Leininger and Anna Lenzhen and Kasra Rafi}, journal={arXiv: Geometric Topology}, year={2013} }
We describe a method for constructing Teichm\"uller geodesics where the vertical measured foliation $\nu$ is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichm\"uller 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 set of all possible measured foliations in the homotopy class of…
23 Citations
Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II
- Mathematics
- 2016
Abstract Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is…
Limits of Teichmüller geodesics in the Universal Teichmüller space
- Mathematics
- 2015
A Thurston boundary of the universal Teichmüller space T(D) is the set of projective bounded measured laminations PMLbdd(D) of D . We prove that each Teichmüller geodesic ray in T(D) converges to a…
MINIMAL NONUNIQUELY ERGODIC VERTICAL FOLIATION, II
- Mathematics
- 2016
Given a sequence of curves on a surface, we provide con- ditions which ensure that (1) the sequence is an innite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is repre-…
A Thurston boundary and visual sphere of the universal Teichmüller space
- MathematicsJournal d'Analyse Mathématique
- 2021
A Thurston boundary of the universal Teichmuller space T(ⅅ) is the space PMLbdd(ⅅ) of projective bounded measured laminations of ⅅ. A geodesic ray in T(ⅅ) is of generalized Teichmuller-type if it…
Limits in 𝒫ℳℱ of Teichmüller geodesics
- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2019
In this paper we consider the limit set in Thurston’s compactification{\mathcal{P}\kern-2.27622pt\mathcal{M}\kern-0.284528pt\mathcal{F}}of Teichmüller space of some Teichmüller geodesics defined by…
On the geometry of the Thurston metric on Teichmüller spaces : geodesics that disobey an analogue of Masur ’ s criterion
- Mathematics
- 2021
We construct a counterexample for an analogue of Masur’s criterion in the setting of Teichmüller space with the Thurston metric. For that, we find a minimal, filling, non-uniquely ergodic lamination…
Exotic limit sets of Teichmüller geodesics in the HHS boundary
- MathematicsGroups, Geometry, and Dynamics
- 2019
We answer a question of Durham, Hagen, and Sisto, proving that a Teichmuller geodesic ray does not necessarily converge to a unique point in the hierarchically hyperbolic space boundary of…
Masur's criterion does not hold in Thurston metric
- Mathematics
- 2019
We construct a counterexample for Masur's criterion in the setting of Teichm\"uller space with Thurston metric. For that, we find a minimal, non-uniquely ergodic lamination $\lambda$ on a seven-times…
Limit sets of Weil-Petersson geodesics
- Mathematics
- 2016
In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichm\"uller space. On the…
Visual sphere and Thurston's boundary of the Universal Teichm\"uller space
- Mathematics
- 2015
Thurston's boundary to the universal Teichm\"uller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in…
References
SHOWING 1-10 OF 69 REFERENCES
Limits in 𝒫ℳℱ of Teichmüller geodesics
- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2019
In this paper we consider the limit set in Thurston’s compactification{\mathcal{P}\kern-2.27622pt\mathcal{M}\kern-0.284528pt\mathcal{F}}of Teichmüller space of some Teichmüller geodesics defined by…
Teichmüller geodesics that do not have a limit in ℳℱ
- Mathematics
- 2005
In this paper we consider a problem in Teichmuller geometry at infinity. Recall that Teichmuller space Tg of a closed oriented surface of genus g equipped with Teichmuller metric is a complete…
Extremal length estimates and product regions in Teichm
- Mathematics
- 1994
We study the Teichm\"uller metric on the Teichm\"uller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the…
Prescribing the behavior of Weil-Petersson geodesics in the moduli space of Riemann surfaces
- Mathematics
- 2012
We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space…
Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows
- Mathematics
- 2008
We define an ending lamination for a Weil–Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil–Petersson metric [Bro2], these ending laminations provide an effective…
Hyperbolicity in Teichmüller space
- Mathematics
- 2014
We give an inductive description of a Teichmuller geodesic, that is, we show that there is a sense in which a Teichmuller geodesic is assembled from Teichmuller geodesics in smaller subsurfaces. We…
A characterization of short curves of a Teichmuller geodesic
- Mathematics
- 2005
We provide a combinatorial condition characterizing curves that are short along a Teichmuller geodesic. This condition is closely related to the condition pro- vided by Minsky for curves in a…
Bounded combinatorics and the Lipschitz metric on Teichmüller space
- Mathematics
- 2010
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…
Lines of Minima and Teichmüller Geodesics
- Mathematics
- 2006
Abstract.For two measured laminations ν+ and ν− that fill up a hyperbolizable surface S and for $$t\,\in\,(-\infty,\infty)$$, let $${\mathcal{L}}_t$$ be the unique hyperbolic surface that minimizes…
On the ergodicity of flat surfaces of finite area
- Mathematics
- 2012
We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmüller orbits are recurrent to a compact set of…