Ergodic invariant measure on the space of geodesic currents

Viveka Erlandsson, Gabriele Mondello

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Abstract

Let S be a compact, connected, oriented surface, possibly with boundary,
of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani’s
and Hamenst¨adt’s classification of (homogeneous and non-homogenous) locally finite mapping class group invariant ergodic measures on the space of measured laminations ML(S) to the space of geodesic currents C(S). Moreover, we also extend Lindenstrauss-Mirzakhani’s classification of orbit closures toC(S). Our argument relies on their results and on the decomposition of a current into a sum of three currents with isotopically disjoint supports: a measured lamination without closed leaves, a simple multi-curve and a current that binds its hull.
Original languageEnglish
Pages (from-to)2449-2513
Number of pages65
JournalAnnales de l'institut Fourier
Volume72
Issue number6
Early online date29 Jul 2022
DOIs
Publication statusPublished - 21 Oct 2022

Bibliographical note

Funding Information:
V.E. was partially supported by Academy of Finland project #297258 and EP-SRC grant EP/T015926/1. G.M. was partially supported by INdAM GNSAGA research group. While writing up the present article, we learnt that a proof of the dichotomy for currents of full hull (Theorem F) was independently obtained by Burger–Iozzi–Parreau–Pozzetti in [4]. The authors would like to thank François Labourie and Chris Leininger for interesting conversation on the topics of this paper. We are particularly indebted to Juan Souto for very valuable feedback, remarks and suggestions. We are also grateful to Marc Burger, Alessandra Iozzi, Anne Parreau and Beatrice Pozzetti for a very useful exchange of ideas after both our preprints had appeared and for spotting some mistakes and incongruences in the first version of this work. We also thank an anonymous referee for carefully reading our paper and for useful comments.

Funding Information:
Keywords: Hyperbolic surfaces, geodesic currents, mapping class group, measure classification. 2020 Mathematics Subject Classification: 20F34, 30F60, 37A05, 57M50, 57S05. (*) V.E. was partially supported by Academy of Finland project #297258 and EP-SRC grant EP/T015926/1. G.M. was partially supported by INdAM GNSAGA research group.

Publisher Copyright:
© 2022 Association des Annales de l'Institut Fourier. All rights reserved.

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