Ergodic invariant measures on the space of geodesic currents
Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2449-2513.

Let S be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss–Mirzakhani’s and Hamenstädt’s classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations (S) to the space of geodesic currents 𝒞(S), and we discuss the homogeneous case. Moreover, we extend Lindenstrauss–Mirzakhani’s classification of orbit closures to 𝒞(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.

Soit S une surface compacte, connexe, orientée, éventuellement à bord, de caractéristique d’Euler négative. Dans cet article nous étendons la classification des mesures ergodiques, localement finies et invariantes sous l’action du mapping class group, sur l’espace des laminations mesurées (S) obtenue par Lindenstrauss–Mirzakhani et Hamenstädt, à l’espace des courants géodésiques 𝒞(S), et nous discutons le cas homogène. De plus, nous étendons la classification de la fermeture des orbites obtenue par Lindenstrauss–Mirzakhani à 𝒞(S). Notre argument repose sur leurs résultats et sur le décomposition d’un courant en une somme de trois courants avec supports isotopiquement disjoints : une lamnation mesurée sans feuilles fermées, une multi-courbe simple et un courant qui remplit son enveloppe.

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DOI: 10.5802/aif.3498
Classification: 20F34,  30F60,  37A05,  57M50,  57S05
Keywords: Hyperbolic surfaces, geodesic currents, mapping class group, measure classification
Erlandsson, Viveka 1; Mondello, Gabriele 2

1 School of Mathematics University of Bristol and Department of Mathematics and Statistics UiT Arctic University of Norway (Norway)
2 Department of Mathematics “Sapienza” Università di Roma (Italy)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Erlandsson, Viveka; Mondello, Gabriele. Ergodic invariant measures on the space of geodesic currents. Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2449-2513. doi : 10.5802/aif.3498. https://aif.centre-mersenne.org/articles/10.5802/aif.3498/

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