no. 6
Ergodic invariant measures on the space of geodesic currents
[Mesures ergodiques invariantes sur l’espace des courants géodésiques]
Erlandsson, Viveka1 ; Mondello, Gabriele2
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)
Annales de l'Institut Fourier, Tome 72 (2022) no. 6, pp. 2449-2513.

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 ℳℒ(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.

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 ℳℒ(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.

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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
Mot clés : Surfaces hyperboliques, courants géodésiques, groupe modulaire, classification de mesures

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)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Erlandsson, Viveka; Mondello, Gabriele. Ergodic invariant measures on the space of geodesic currents. Annales de l'Institut Fourier, Tome 72 (2022) no. 6, pp. 2449-2513. doi : 10.5802/aif.3498. https://aif.centre-mersenne.org/articles/10.5802/aif.3498/

[1] Bankovic, Anja; Leininger, Christopher J. Marked-length-spectral rigidity for flat metrics, Trans. Am. Math. Soc., Volume 370 (2018) no. 3, pp. 1867-1884

[2] Bonahon, Francis Bouts des variétés hyperboliques de dimension 3, Ann. Math., Volume 124 (1986) no. 1, pp. 71-158 | DOI | MR

[3] Bonahon, Francis The geometry of Teichmüller space via geodesic currents, Invent. Math., Volume 92 (1988) no. 1, pp. 139-162

[4] Burger, Marc; Iozzi, Alessandra; Perreau, Anne; Pozzetti, Maria Beatrice A structure theorem for geodesic currents and length spectrum compactifications (2017) | arXiv

[5] Burger, Marc; Iozzi, Alessandra; Perreau, Anne; Pozzetti, Maria Beatrice Currents, systoles, and compactifications of character varieties (2019) | arXiv

[6] Constantine, David Marked length spectrum rigidity in nonpositive curvature with singularities, Indiana Univ. Math. J., Volume 67 (2018), pp. 2337-2361

[7] Duchin, Moon; Leininger, Christopher J.; Rafi, Kasra Length spectra and degeneration of flat metrics, Invent. Math., Volume 182 (2010) no. 2, pp. 231-277

[8] Erlandsson, Viveka; Parlier, Hugo; Souto, Juan Counting curves, and the stable length of currents, J. Eur. Math. Soc., Volume 22 (2020) no. 6, pp. 1675-1702 | DOI | MR

[9] Erlandsson, Viveka; Souto, Juan Counting curves in hyperbolic surfaces, Geom. Funct. Anal., Volume 26 (2016) no. 3, pp. 729-777

[10] Fathi, Albert; Laudenbach, François; Poénaru, Valentin Travaux de Thurston sur les surfaces, Astérisque, 66, Société Mathématique de France, 1979, 284 pages (Séminaire Orsay, with an English summary) | MR

[11] Glorieux, Olivier Critical exponent for geodesic currents (2017) | arXiv

[12] Hamenstädt, Ursula Invariant Radon measures on measured lamination space, Invent. Math., Volume 176 (2009) no. 2, pp. 223-273

[13] Ivanov, Nikolai V. Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115, American Mathematical Society, 1992, xii+127 pages (Translated from the Russian by E. J. F. Primrose and revised by the author) | DOI | MR

[14] Levitt, Gilbert Foliations and laminations on hyperbolic surfaces, Topology, Volume 22 (1983) no. 2, pp. 119-135 | DOI | MR

[15] Lindenstrauss, Elon; Mirzakhani, Maryam Ergodic theory of the space of measured laminations, Int. Math. Res. Not. (2008) no. 4, rnm126, 49 pages

[16] Masur, Howard Ergodic actions of the mapping class group, Proc. Am. Math. Soc., Volume 94 (1985) no. 3, pp. 455-459

[17] Mirzakhani, Maryam Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. Math., Volume 168 (2008) no. 1, pp. 97-125

[18] Mirzakhani, Maryam Counting mapping class group orbits on hyperbolic surfaces (2016) | arXiv

[19] Monin, Leonid; Telpukhovskiy, Vania On normalizations of Thurston measure on the space of measured laminations, Topology Appl., Volume 267 (2019), p. 106878 http://www.sciencedirect.com/science/article/pii/s0166864119302925 | DOI

[20] Otal, Jean-Pierre Le spectre marqué des longueurs des surfaces à courbure négative, Ann. Math., Volume 131 (1990) no. 1, pp. 151-162

[21] Penner, Robert C.; Harer, John L. Combinatorics of train tracks, Annals of Mathematics Studies, 125, Princeton University Press, 1992, xii+216 pages | Zbl

[22] Rafi, Kasra; Souto, Juan Geodesic currents and counting problems, Geom. Funct. Anal., Volume 29 (2019) no. 3, pp. 871-889 | DOI | MR

[23] Rivin, Igor Simple curves on surfaces, Geom. Dedicata, Volume 87 (2001) no. 1-3, pp. 345-360 | DOI | MR

[24] Sapir, Jenya Bounds on the number of non-simple closed geodesics on a surface, Int. Math. Res. Not. (2016) no. 24, pp. 7499-7545

[25] Thurston, William P. The Geometry and Topology of Three-manifolds (1980) (unpublished notes, available at http://library.msri.org/books/gt3m/PDF/Thurston-gt3m.pdf)

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