# Recurrent Weil-Petersson geodesic rays with non-uniquely ergodic ending laminations

@article{Brock2014RecurrentWG, title={Recurrent Weil-Petersson geodesic rays with non-uniquely ergodic ending laminations}, author={Jeffrey F. Brock and Babak Modami}, journal={arXiv: Geometric Topology}, year={2014} }

We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichm\"uller geodesics does not hold for WP geodesics.

## 10 Citations

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…

Limit sets of Weil–Petersson geodesics with nonminimal ending laminations

- MathematicsJournal of Topology and Analysis
- 2018

In this paper, we construct examples of Weil–Petersson geodesics with nonminimal ending laminations which have [Formula: see text]-dimensional limit sets in the Thurston compactification of…

ITERATION OF MAPPING CLASSES AND LIMITS OF WEIL-PETERSSON GEODESICS

- Mathematics
- 2015

Let S = Sg be a closed surface of genus g with g > 2, Mod(S) be the mapping class group of S and Teich(S) be the Teichmüller space of S endowed with the Weil-Petersson metric. Fix X,Y ∈ Teich(S). In…

Limit sets of Teichm\"uller geodesics with minimal non-uniquely ergodic vertical foliation

- Mathematics
- 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…

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…

Teichmueller flow and Weil-Petersson flow

- Mathematics
- 2015

For a non-exceptional oriented surface S let Q(S) be the moduli space of area one quadratic differentials. We show that there is a Borel subset E of Q(S) which is invariant under the Teichmueller…

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…

Growth of the Weil–Petersson inradius of moduli space

- MathematicsAnnales de l'Institut Fourier
- 2019

In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichm\"uller space endowed with the…

Weil--Petersson geodesics on the modular surface

- Mathematics
- 2020

We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the…

## References

SHOWING 1-10 OF 41 REFERENCES

Asymptotics of Weil–Petersson Geodesics II: Bounded Geometry and Unbounded Entropy

- Mathematics
- 2010

We use ending laminations for Weil–Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil–Petersson geodesic segments, rays, and lines. Further, a more…

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…

Geometry of the Weil-Petersson completion of Teichm\

- Mathematics
- 2003

We present a view of the current understanding of the geometry of Weil-Petersson (WP) geodesics on the completion of the Teichm\"uller space. We sketch a collection of results by other authors and…

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…

Classification of Weil-Petersson isometries

- Mathematics
- 2002

This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmüller space for any choice of pseudo-Anosov mapping class. As a consequence one…

Almost filling laminations and the connectivity of ending lamination space

- Mathematics
- 2008

We show that if S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then the ending lamination space of S is…

Limit sets of Teichm\"uller geodesics with minimal non-uniquely ergodic vertical foliation

- Mathematics
- 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…

Covers and curve complex

- Mathematics
- 2007

We provide the first nontrivial examples of quasi-isometric embeddings between curve complexes; these are induced by orbifold covers. This leads to new quasi-isometric embeddings between mapping…

Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity

- Mathematics
- 2008

We analyze the coarse geometry of the Weil-Petersson metric on Teichm¨ uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the…