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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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1435-5345
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Volume 2019, Issue 747

# Limits in 𝒫ℳℱ of Teichmüller geodesics

Jon Chaika
/ Howard Masur
/ Michael Wolf
Published Online: 2016-07-12 | DOI: https://doi.org/10.1515/crelle-2016-0017

## Abstract

In this paper we consider the limit set in Thurston’s compactification $\mathcal{𝒫}\mathcal{ℳ}\mathcal{ℱ}$ of Teichmüller space of some Teichmüller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that (a) there are quadratic differentials so that the limit set of the geodesic is a unique point, (b) there are quadratic differentials so that the limit set is a line segment, (c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and (d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmüller geodesics whose limit sets overlap and Teichmüller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmüller geodesic and a simple closed curve γ so that the hyperbolic length of the geodesic in the homotopy class of γ varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.

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Revised: 2015-10-19

Published Online: 2016-07-12

Published in Print: 2019-02-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1107452

Award identifier / Grant number: DMS-1107263

Award identifier / Grant number: DMS-1107367

Award identifier / Grant number: DMS-1004372

Award identifier / Grant number: DMS-1300550

Award identifier / Grant number: DMS-0905907

Award identifier / Grant number: DMS-1205016

Award identifier / Grant number: DMS-1007383

Research of Jon Chaika partially supported by DMS-1004372 and DMS-1300550. Research of Howard Masur partially supported by DMS-0905907 and DMS-1205016. Research of Michael Wolf partially supported by DMS-1007383 and the Morningside Center (Tsinghua University). Howard Masur and Michael Wolf appreciate the support of the GEAR Network (DMS-1107452, DMS-1107263, DMS-1107367).

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 747, Pages 1–44, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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