Let 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 to the space of geodesic currents , and we discuss the homogeneous case. Moreover, we extend Lindenstrauss–Mirzakhani’s classification of orbit closures to . 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 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 obtenue par Lindenstrauss–Mirzakhani et Hamenstädt, à l’espace des courants géodésiques , et nous discutons le cas homogène. De plus, nous étendons la classification de la fermeture des orbites obtenue par Lindenstrauss–Mirzakhani à . 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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Accepted:
Published online:
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
@article{AIF_2022__72_6_2449_0, author = {Erlandsson, Viveka and Mondello, Gabriele}, title = {Ergodic invariant measures on the space of geodesic currents}, journal = {Annales de l'Institut Fourier}, pages = {2449--2513}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {72}, number = {6}, year = {2022}, doi = {10.5802/aif.3498}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3498/} }
TY - JOUR AU - Erlandsson, Viveka AU - Mondello, Gabriele TI - Ergodic invariant measures on the space of geodesic currents JO - Annales de l'Institut Fourier PY - 2022 SP - 2449 EP - 2513 VL - 72 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3498/ DO - 10.5802/aif.3498 LA - en ID - AIF_2022__72_6_2449_0 ER -
%0 Journal Article %A Erlandsson, Viveka %A Mondello, Gabriele %T Ergodic invariant measures on the space of geodesic currents %J Annales de l'Institut Fourier %D 2022 %P 2449-2513 %V 72 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3498/ %R 10.5802/aif.3498 %G en %F AIF_2022__72_6_2449_0
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/
[1] Marked-length-spectral rigidity for flat metrics, Trans. Am. Math. Soc., Volume 370 (2018) no. 3, pp. 1867-1884
[2] Bouts des variétés hyperboliques de dimension , Ann. Math., Volume 124 (1986) no. 1, pp. 71-158 | DOI | MR
[3] The geometry of Teichmüller space via geodesic currents, Invent. Math., Volume 92 (1988) no. 1, pp. 139-162
[4] A structure theorem for geodesic currents and length spectrum compactifications (2017) | arXiv
[5] Currents, systoles, and compactifications of character varieties (2019) | arXiv
[6] Marked length spectrum rigidity in nonpositive curvature with singularities, Indiana Univ. Math. J., Volume 67 (2018), pp. 2337-2361
[7] Length spectra and degeneration of flat metrics, Invent. Math., Volume 182 (2010) no. 2, pp. 231-277
[8] Counting curves, and the stable length of currents, J. Eur. Math. Soc., Volume 22 (2020) no. 6, pp. 1675-1702 | DOI | MR
[9] Counting curves in hyperbolic surfaces, Geom. Funct. Anal., Volume 26 (2016) no. 3, pp. 729-777
[10] Travaux de Thurston sur les surfaces, Astérisque, 66, Société Mathématique de France, 1979, 284 pages (Séminaire Orsay, with an English summary) | MR
[11] Critical exponent for geodesic currents (2017) | arXiv
[12] Invariant Radon measures on measured lamination space, Invent. Math., Volume 176 (2009) no. 2, pp. 223-273
[13] 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] Foliations and laminations on hyperbolic surfaces, Topology, Volume 22 (1983) no. 2, pp. 119-135 | DOI | MR
[15] Ergodic theory of the space of measured laminations, Int. Math. Res. Not. (2008) no. 4, rnm126, 49 pages
[16] Ergodic actions of the mapping class group, Proc. Am. Math. Soc., Volume 94 (1985) no. 3, pp. 455-459
[17] Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. Math., Volume 168 (2008) no. 1, pp. 97-125
[18] Counting mapping class group orbits on hyperbolic surfaces (2016) | arXiv
[19] 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] Le spectre marqué des longueurs des surfaces à courbure négative, Ann. Math., Volume 131 (1990) no. 1, pp. 151-162
[21] Combinatorics of train tracks, Annals of Mathematics Studies, 125, Princeton University Press, 1992, xii+216 pages | Zbl
[22] Geodesic currents and counting problems, Geom. Funct. Anal., Volume 29 (2019) no. 3, pp. 871-889 | DOI | MR
[23] Simple curves on surfaces, Geom. Dedicata, Volume 87 (2001) no. 1-3, pp. 345-360 | DOI | MR
[24] Bounds on the number of non-simple closed geodesics on a surface, Int. Math. Res. Not. (2016) no. 24, pp. 7499-7545
[25] 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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