Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space

Eskin, Alex; Mirzakhani, Maryam

doi:10.1007/s10240-018-0099-2

Invariant and stationary measures for the

SL (2, ℝ)

action on Moduli space

Eskin, Alex ; Mirzakhani, Maryam

Publications Mathématiques de l'IHÉS, Tome 127 (2018), pp. 95-324.

Résumé

We prove some ergodic-theoretic rigidity properties of the action of $SL (2, ℝ)$ on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of $SL (2, ℝ)$ is supported on an invariant affine submanifold.

The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.

| 1 citation dans Numdam

DOI : 10.1007/s10240-018-0099-2

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     title = {Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on {Moduli} space},
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Eskin, Alex; Mirzakhani, Maryam. Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space. Publications Mathématiques de l'IHÉS, Tome 127 (2018), pp. 95-324. doi : 10.1007/s10240-018-0099-2. https://www.numdam.org/articles/10.1007/s10240-018-0099-2/

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Lindenstrauss, Elon; Mohammadi, Amir; Wang, Zhiren Polynomial effective equidistribution, Comptes Rendus. Mathématique, Volume 361 (2023) no. G2, p. 507 | DOI:10.5802/crmath.411
Chen, Dawei; Möller, Martin; Sauvaget, Adrien Masur–Veech volumes and intersection theory: The principal strata of quadratic differentials, Duke Mathematical Journal, Volume 172 (2023) no. 9 | DOI:10.1215/00127094-2022-0063
Prohaska, Roland; Sert, Cagri; Shi, Ronggang Expanding measures: Random walks and rigidity on homogeneous spaces, Forum of Mathematics, Sigma, Volume 11 (2023) | DOI:10.1017/fms.2023.56
Calsamiglia, Gabriel; Deroin, Bertrand; Francaviglia, Stefano A transfer principle: from periods to isoperiodic foliations, Geometric and Functional Analysis, Volume 33 (2023) no. 1, p. 57 | DOI:10.1007/s00039-023-00627-w
Dozier, Benjamin Measure bound for translation surfaces with short saddle connections, Geometric and Functional Analysis, Volume 33 (2023) no. 2, p. 421 | DOI:10.1007/s00039-023-00636-9
Katz, Asaf Measure rigidity of Anosov flows via the factorization method, Geometric and Functional Analysis, Volume 33 (2023) no. 2, p. 468 | DOI:10.1007/s00039-023-00629-8
Arana-Herrera, Francisco Bounds on the Hausdorff Dimension of Teichmüller Horocycle Flow Orbit Closures, International Mathematics Research Notices, Volume 2023 (2023) no. 14, p. 12064 | DOI:10.1093/imrn/rnac181
Cantat, Serge; Dujardin, Romain Random dynamics on real and complex projective surfaces, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 0 (2023) no. 0 | DOI:10.1515/crelle-2023-0038
Flake, Johannes; Thevis, Andrea Strata ofp‐origamis, Mathematische Nachrichten, Volume 296 (2023) no. 3, p. 1087 | DOI:10.1002/mana.202100290
Skripchenko, Alexandra Sergeevna Renormalization in one-dimensional dynamics, Russian Mathematical Surveys, Volume 78 (2023) no. 6, p. 983 | DOI:10.4213/rm10110e
Skripchenko, Alexandra Sergeevna Ренормализация в одномерной динамике, Успехи математических наук, Volume 78 (2023) no. 6(474), p. 3 | DOI:10.4213/rm10110
Mullane, Scott Strata of differentials of the second kind, positivity and irreducibility of certain Hurwitz spaces, Annales de l'Institut Fourier, Volume 72 (2022) no. 4, p. 1379 | DOI:10.5802/aif.3497
FORNI, GIOVANNI Twisted cohomological equations for translation flows, Ergodic Theory and Dynamical Systems, Volume 42 (2022) no. 3, p. 881 | DOI:10.1017/etds.2021.120
Lanneau, Erwan; Möller, Martin Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci, Experimental Mathematics, Volume 31 (2022) no. 2, p. 621 | DOI:10.1080/10586458.2019.1671920
Costantini, Matteo; Möller, Martin; Zachhuber, Jonathan The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials, Forum of Mathematics, Pi, Volume 10 (2022) | DOI:10.1017/fmp.2022.10
Dowdall, Spencer; Work, Grace Discretely shrinking targets in moduli space, Geometriae Dedicata, Volume 216 (2022) no. 5 | DOI:10.1007/s10711-022-00716-4
Mohammadi, Amir; Rafi, Kasra Benjamini-Schramm convergence of periodic orbits, Geometriae Dedicata, Volume 216 (2022) no. 5 | DOI:10.1007/s10711-022-00703-9
Apisa, Paul; Saavedra, Rafael M.; Zhang, Christopher Periodic points on the regular and double n-gon surfaces, Geometriae Dedicata, Volume 216 (2022) no. 6 | DOI:10.1007/s10711-022-00730-6
Benirschke, Frederik; Dozier, Benjamin; Grushevsky, Samuel Equations of linear subvarieties of strata of differentials, Geometry Topology, Volume 26 (2022) no. 6, p. 2773 | DOI:10.2140/gt.2022.26.2773
Ygouf, Florent Isoperiodic Dynamics in Rank 1 Affine Invariant Orbifolds, International Mathematics Research Notices, Volume 2022 (2022) no. 24, p. 19854 | DOI:10.1093/imrn/rnab153
Marmi, Stefano Some arithmetical aspects of renormalization in Teichmüller dynamics: On the occasion of Corinna Ulcigrai winning the Brin Prize, Journal of Modern Dynamics, Volume 18 (2022) no. 0, p. 131 | DOI:10.3934/jmd.2022006
Kahn, Jeremy; Wright, Alex Hodge and Teichmüller, Journal of Modern Dynamics, Volume 18 (2022) no. 0, p. 149 | DOI:10.3934/jmd.2022007
Bainbridge, Matt; Smillie, John; Weiss, Barak Horocycle Dynamics: New Invariants and Eigenform Loci in the Stratum ℋ(1,1), Memoirs of the American Mathematical Society, Volume 280 (2022) no. 1384 | DOI:10.1090/memo/1384
Massart, Daniel A Short Introduction to Translation Surfaces, Veech Surfaces, and Teichmüller Dynamics, Surveys in Geometry I (2022), p. 343 | DOI:10.1007/978-3-030-86695-2_9
FORNI, GIOVANNI Sobolev regularity of solutions of the cohomological equation, Ergodic Theory and Dynamical Systems, Volume 41 (2021) no. 3, p. 685 | DOI:10.1017/etds.2019.108
FORNI, GIOVANNI Limits of geodesic push-forwards of horocycle invariant measures, Ergodic Theory and Dynamical Systems, Volume 41 (2021) no. 9, p. 2782 | DOI:10.1017/etds.2020.87
Apisa, Paul; Wright, Alex Marked points on translation surfaces, Geometry Topology, Volume 25 (2021) no. 6, p. 2913 | DOI:10.2140/gt.2021.25.2913
Prohaska, Roland On convergence of random walks on moduli space, Illinois Journal of Mathematics, Volume 65 (2021) no. 3 | DOI:10.1215/00192082-9421088
Koziarz, Vincent; Nguyen, Duc-Manh Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares, Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), p. 831 | DOI:10.5802/jep.159
Chen, Dawei; Wright, Alex The WYSIWYG compactification, Journal of the London Mathematical Society, Volume 103 (2021) no. 2, p. 490 | DOI:10.1112/jlms.12382
Grechuk, Bogdan Analysis, Landscape of 21st Century Mathematics (2021), p. 103 | DOI:10.1007/978-3-030-80627-9_3
Nevo, Amos; Rühr, Rene; Weiss, Barak Effective counting on translation surfaces, Advances in Mathematics, Volume 360 (2020), p. 106890 | DOI:10.1016/j.aim.2019.106890
Wright, Alex A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces, Bulletin of the American Mathematical Society, Volume 57 (2020) no. 3, p. 359 | DOI:10.1090/bull/1687
DELECROIX, VINCENT; GOUJARD, ÉLISE; ZOGRAF, PETER; ZORICH, ANTON ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES, Forum of Mathematics, Pi, Volume 8 (2020) | DOI:10.1017/fmp.2020.2
Apisa, Paul GL2ℝ–invariant measures in marked strata : generic marked points, Earle–Kra for strata and illumination, Geometry Topology, Volume 24 (2020) no. 1, p. 373 | DOI:10.2140/gt.2020.24.373
Pardo, Angel A Non-varying Phenomenon with an Application to the Wind-Tree Model, International Mathematics Research Notices, Volume 2020 (2020) no. 18, p. 5642 | DOI:10.1093/imrn/rny188
Chen, Dawei; Möller, Martin; Sauvaget, Adrien; Zagier, Don Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials, Inventiones mathematicae, Volume 222 (2020) no. 1, p. 283 | DOI:10.1007/s00222-020-00969-4
Nguyen, Duc-Manh; Pan, Huiping; Su, Weixu Existence of closed geodesics through a regular point on translation surfaces, Mathematische Annalen, Volume 376 (2020) no. 1-2, p. 583 | DOI:10.1007/s00208-019-01897-2
Chen, Dawei Positivity of Divisor Classes on the Strata of Differentials, The Quarterly Journal of Mathematics, Volume 71 (2020) no. 1, p. 247 | DOI:10.1093/qmath/haz032
Magee, Michael On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials, Compositio Mathematica, Volume 155 (2019) no. 12, p. 2354 | DOI:10.1112/s0010437x1900767x
Apisa, Paul Rank One Orbit Closures in $\varvec H^{\varvec \lowercase h y p} (\varvec \lowercase g - 1, \varvec \lowercase g - 1)$ , Geometric and Functional Analysis, Volume 29 (2019) no. 6, p. 1617 | DOI:10.1007/s00039-019-00513-4
Eskin, Alex; Matheus, Carlos Semisimplicity of the Lyapunov spectrum for irreducible cocycles, Israel Journal of Mathematics, Volume 230 (2019) no. 2, p. 973 | DOI:10.1007/s11856-019-1841-2
Frączek, Krzysztof Ergodic properties of the ideal gas model for infinite billiards, Physica D: Nonlinear Phenomena, Volume 390 (2019), p. 9 | DOI:10.1016/j.physd.2018.10.004
Chen, Dawei; Chen, Qile Spin and hyperelliptic structures of log twisted differentials, Selecta Mathematica, Volume 25 (2019) no. 2 | DOI:10.1007/s00029-019-0467-x
Eskin, Alex; Filip, Simion; Wright, Alex The algebraic hull of the Kontsevich–Zorich cocycle, Annals of Mathematics, Volume 188 (2018) no. 1 | DOI:10.4007/annals.2018.188.1.5
Frączek, Krzysztof; Schmoll, Martin On Ergodicity of Foliations on $Z^{d}$ Z d -Covers of Half-Translation Surfaces and Some Applications to Periodic Systems of Eaton Lenses, Communications in Mathematical Physics, Volume 362 (2018) no. 2, p. 609 | DOI:10.1007/s00220-018-3186-9
Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin Compactification of strata of Abelian differentials, Duke Mathematical Journal, Volume 167 (2018) no. 12 | DOI:10.1215/00127094-2018-0012

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