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

Jeffrey Brock EMAIL logo , Christopher Leininger , Babak Modami and Kasra Rafi


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 represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension 3g-3, for example.

We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex.

Award Identifier / Grant number: DMS-1207572

Award Identifier / Grant number: DMS-1510034

Award Identifier / Grant number: DMS-1065872

Award Identifier / Grant number: 435885

Funding statement: The first author was partially supported by NSF grant DMS-1207572, the second author by NSF grant DMS-1510034, the third author by NSF grant DMS-1065872 and the fourth author by NSERC grant # 435885.


We would like to thank Howard Masur for illuminating conversations and communications as well as the anonymous referee for helpful suggestions. We also would like to thank Anna Lenzhen; her collaboration in the first paper was crucial for the development of the current paper. Finally, we would like to thank MSRI at Berkeley for hosting the program Dynamics on moduli spaces in April 2015; where the authors had the chance to form some of the techniques of this paper.


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Received: 2016-04-19
Revised: 2017-02-17
Published Online: 2017-06-07
Published in Print: 2020-01-01

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