Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



From rational billiards to dynamics on moduli spaces

Author: Alex Wright
Journal: Bull. Amer. Math. Soc. 53 (2016), 41-56
MSC (2010): Primary 22E60, 15A57, 17B20, 58C35
Published electronically: September 8, 2015
MathSciNet review: 3403080
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces and, in particular, the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi. We also discuss the context and applications of this result, and its connections to other areas of mathematics, such as algebraic geometry, Teichmüller theory, and ergodic theory on homogeneous spaces.

References [Enhancements On Off] (What's this?)

  • [ANW] David Aulicino, Duc-Manh Nguyen, and Alex Wright, Classification of higher rank orbit closures in $ \mathcal {H}^{\textrm {odd}(4)}$, preprint, arXiv 1308.5879 (2013), to appear in Geom. Top.
  • [BQ09] Yves Benoist and Jean-François Quint, Mesures stationnaires et fermés invariants des espaces homogènes, C. R. Math. Acad. Sci. Paris 347 (2009), no. 1-2, 9-13 (French, with English and French summaries). MR 2536741 (2010g:60014),
  • [BRH] Aaron Brown and Federico Rodriguez-Hertz, Measure rigidity for random dynamics on surfaces and related skew products, preprint, arXiv 1406.7201 (2014).
  • [CE] Jon Chaika and Alex Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, preprint, arXiv 1305.1104 (2014).
  • [DHL14] Vincent Delecroix, Pascal Hubert, and Samuel Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 6, 1085-1110 (English, with English and French summaries). MR 3297155
  • [DM90] S. G. Dani and G. A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces of $ {\rm SL}(3,{\bf R})$, Math. Ann. 286 (1990), no. 1-3, 101-128. MR 1032925 (91k:22026),
  • [Ein06] Manfred Einsiedler, Ratner's theorem on $ {\rm SL}(2,\mathbb{R})$-invariant measures, Jahresber. Deutsch. Math.-Verein. 108 (2006), no. 3, 143-164. MR 2265534 (2008b:37048)
  • [Ein09] Manfred Einsiedler, What is $ \dots $ measure rigidity?, Notices Amer. Math. Soc. 56 (2009), no. 5, 600-601. MR 2509063 (2010j:37004)
  • [EKL06] Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. (2) 164 (2006), no. 2, 513-560. MR 2247967 (2007j:22032),
  • [EM] Alex Eskin and Maryam Mirzakhani, Invariant and stationary measures for the $ SL(2,\mathbb{R})$ action on moduli space, preprint, arXiv 1302.3320 (2013).
  • [EMM] Alex Eskin, Maryam Mirzakhani, and Amir Mohammadi, Isolation theorems for $ SL(2, \mathbb{R})$-invariant submanifolds in moduli space, preprint, arXiv:1305.3015 (2013).
  • [Esk] Alex Eskin, Lectures on the $ {SL}(2, \mathbb{R})$ action on moduli space,
  • [Fila] Simion Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, preprint, arXiv:1307.7314 (2013).
  • [Filb] Simion Filip, Splitting mixed Hodge structures over affine invariant manifolds, preprint, arXiv:1311.2350 (2013).
  • [For02] Giovanni Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), no. 1, 1-103. MR 1888794 (2003g:37009),
  • [KMS86] Steven Kerckhoff, Howard Masur, and John Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2, 293-311. MR 855297 (88f:58122),
  • [Kon97] M. Kontsevich, Lyapunov exponents and Hodge theory, The mathematical beauty of physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 318-332. MR 1490861 (99b:58147)
  • [KZ03] Maxim Kontsevich and Anton Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), no. 3, 631-678. MR 2000471 (2005b:32030),
  • [LMW] Samuel Lelièver, Thierry Monteil, and Barak Weiss, Everything is illuminated, preprint, arXiv 1407.2975 (2014).
  • [Mas82] Howard Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169-200. MR 644018 (83e:28012),
  • [Mas88] Howard Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 215-228. MR 955824 (90e:30046),
  • [Mas90] Howard Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 151-176. MR 1053805 (91d:30042),
  • [Mas92] Howard Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), no. 3, 387-442. MR 1167101 (93f:30045),
  • [McM03] Curtis T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857-885 (electronic). MR 1992827 (2004f:32015),
  • [McM14] Curtis T. McMullen, The evolution of geometric structures on 3-manifolds, The Poincaré conjecture, Clay Math. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 2014, pp. 31-46. MR 3308757
  • [Möl06a] Martin Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math. 165 (2006), no. 3, 633-649. MR 2242629 (2007e:14012),
  • [Möl06b] Martin Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), no. 2, 327-344. MR 2188128 (2007b:32026),
  • [Mor05] Dave Witte Morris, Ratner's theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005. MR 2158954 (2006h:37006)
  • [MT94] G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), no. 1-3, 347-392. MR 1253197 (95k:22013),
  • [MT02] Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089. MR 1928530 (2003j:37002),
  • [NW14] Duc-Manh Nguyen and Alex Wright, Non-Veech surfaces in $ \mathcal {H}^{\rm hyp}(4)$ are generic, Geom. Funct. Anal. 24 (2014), no. 4, 1316-1335. MR 3248487,
  • [Vee82] William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201-242. MR 644019 (83g:28036b),
  • [Ven08] Akshay Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 117-134. MR 2358379 (2009d:11105),
  • [Wri14] Alex Wright, The field of definition of affine invariant submanifolds of the moduli space of abelian differentials, Geom. Topol. 18 (2014), no. 3, 1323-1341. MR 3254934,
  • [Wri15a] Alex Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015), no. 1, 413-438. MR 3318755,
  • [Wri15b] Alex Wright, Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci. 2 (2015), no. 1, 63-108. MR 3354955,
  • [Zor] Anton Zorich, The magic wand theorem of A. Eskin and M. Mirzakhani, English translation of paper in Gaz. Math., arXiv 1502.05654 (2015).
  • [Zor06] Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583. MR 2261104 (2007i:37070),
  • [Zor14] Anton Zorich, Le théorème de la baguette magique de A. Eskin et M. Mirzakhani, Gaz. Math. (2014), no. 142, 39-54.

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2010): 22E60, 15A57, 17B20, 58C35

Retrieve articles in all journals with MSC (2010): 22E60, 15A57, 17B20, 58C35

Additional Information

Alex Wright
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois

Received by editor(s): May 8, 2015
Published electronically: September 8, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society