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Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows

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Abstract

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 boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex \({\mathcal{C}(S)}\). As an application, we establish fundamentals of the topological dynamics of the Weil–Petersson geodesic flow, showing density of closed orbits and topological transitivity.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, Box 1917, Providence, RI, 02912, USA

    Jeffrey Brock

  2. Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, USA

    Howard Masur

  3. Department of Mathematics, Yale University, New Haven, CT, 06520, USA

    Yair Minsky

Authors
  1. Jeffrey Brock
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  2. Howard Masur
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  3. Yair Minsky
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Correspondence to Jeffrey Brock.

Additional information

J.B. was supported by NSF Grant DMS-0505442 and a John S. Guggenheim Foundation Fellowship. H.M. was supported by NSF Grant DMS-0603980. Y.M. was supported by NSF Grant DMS-0504019.

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Brock, J., Masur, H. & Minsky, Y. Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows. Geom. Funct. Anal. 19, 1229–1257 (2010). https://doi.org/10.1007/s00039-009-0034-2

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  • DOI: https://doi.org/10.1007/s00039-009-0034-2

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