Teichmüller geodesics with $ d$-dimensional limit sets

Anna Lenzhen; Babak Modami; Kasra Rafi

doi:10.3934/jmd.2018010

Article Contents

2018, Volume 12: 261-283. Doi: 10.3934/jmd.2018010

This volume Previous Article On the non-equivalence of the Bernoulli and $ K$ properties in dimension four Next Article On manifolds admitting stable type Ⅲ$_{\textbf1}$ Anosov diffeomorphisms

Teichmüller geodesics with $ d$-dimensional limit sets

1.
Laboratoire de Mathématiques, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
2.
Mathematics Department, Stony Brook, University, Stony Brook, NY 11794-3651, USA
3.
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Received: September 03, 2016

Revised: May 06, 2018

Published: August 2018

BM: Partially supported by NSF grant DMS-1065872.
KR: Partially supported by NSERC Discovery grant RGPIN-435885.

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Abstract

We construct an example of a Teichmüller geodesic ray whose limit set in the Thurston boundary of Teichmüller space is a $ d$-dimensional simplex.

Keywords:

Teichmüller geodesics,
Thurston boundary of Teichmüller space,
nonuniquely ergodic measured foliations,
quadratic differentials.

Mathematics Subject Classification: Primary: 32G15; Secondary: 57M50.

Citation:

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Figure 1. Case $d = 2$. The surface $X_0$ is glued out of three tori

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Figure 2. Annuli in $T^i$ about $\beta^i$ and $\alpha_n^i$. Expanding annulus with core curve $\beta^i$, on the left, and flat annulus about $\alpha_n^i$, on the right, at the time when $\alpha_n^i$ is balanced

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Figure 3. The interval when $\alpha_k^i$ is short. For $k = 2n$, the curves $\alpha_k^i,\; i\in\mathbb{Z}_3$, start getting short at different times, but they grow back to length 1 roughly at the same time. We choose $t_n$ in the shaded interval to guarantee that all three curves are short and have collar neighborhoods of approximately the same width

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