# American Institute of Mathematical Sciences

July & October  2014, 8(3&4): 271-436. doi: 10.3934/jmd.2014.8.271

## Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

 1 Department of Mathematics, University of Maryland, College Park, MD 20742-4015 2 CNRS, LAGA, Institut Galilée, Université Paris 13, 99, Av. Jean-Baptiste Clément 93430, Villetaneuse, France

Received  January 2013 Revised  June 2014 Published  April 2015

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Będlewo school Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'' (from 4 to 16 July, 2011).
In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmüller flow and the Kontsevich--Zorich cocycle work as renormalization dynamics for interval exchange transformations and translation flows.
In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).
Citation: Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
##### References:
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Hautes Études Sci., 104 (2006), 143. doi: 10.1007/s10240-006-0001-5. Google Scholar [7] A. Avila, C. Matheus and J.-C. Yoccoz, On the Kontsevich-Zorich cocycle for McMullen's family of symmetric translation surfaces,, in preparation., (). Google Scholar [8] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1. Google Scholar [9] A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications,, Invent. Math., 181 (2010), 115. doi: 10.1007/s00222-010-0243-1. Google Scholar [10] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two,, Geom. Topol., 11 (2007), 1887. doi: 10.2140/gt.2007.11.1887. Google Scholar [11] N. Bergeron, Le Spectre des Surfaces Hyperboliques,, Savoirs Actuels (Les Ulis), (2011). Google Scholar [12] J. Borwein and P. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity,, Reprint of the 1987 original, (1987). Google Scholar [13] K. Burns, H. Masur and A. Wilkinson, The Weil-Petersson geodesic flow is ergodic,, Ann. of Math. (2), 175 (2012), 835. doi: 10.4007/annals.2012.175.2.8. Google Scholar [14] A. Bufetov, Limit theorems for translations flows,, Ann. of Math. (2), 179 (2014), 431. doi: 10.4007/annals.2014.179.2.2. Google Scholar [15] K. Calta, Veech surfaces and complete periodicity in genus two,, J. Amer. Math. Soc., 17 (2004), 871. doi: 10.1090/S0894-0347-04-00461-8. Google Scholar [16] D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying strata,, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 309. Google Scholar [17] Y. Cheung and A. Eskin, Unique ergodicity of translation flows,, in Partially Hyperbolic Dynamics, (2007), 213. Google Scholar [18] V. Delecroix, Cardinality of Rauzy classes,, Ann. Inst. Fourier (Grenoble), 63 (2013), 1651. doi: 10.5802/aif.2811. Google Scholar [19] P. Deligne, Un théorème de finitude pour la monodromie,, in Discrete Groups in Geometry and Analysis (New Haven, (1984), 1. doi: 10.1007/978-1-4899-6664-3_1. Google Scholar [20] V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, , (2011), 1. Google Scholar [21] V. Delecroix and C. Matheus, Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich,, , (2011), 1. Google Scholar [22] , Disquisitiones Mathematicae,, , (). Google Scholar [23] J. Ellenberg and D. B. McReynolds, Arithmetic Veech sublattices of $SL(2,\mathbbZ)$,, Duke Math. J., 161 (2012), 415. doi: 10.1215/00127094-1507412. Google Scholar [24] A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, Publ. Math. Inst. Hautes Études Sci., 120 (2014), 207. doi: 10.1007/s10240-013-0060-3. Google Scholar [25] A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319. Google Scholar [26] A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants,, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61. doi: 10.1007/s10240-003-0015-1. Google Scholar [27] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbbR)$ action on moduli space,, , (). Google Scholar [28] A. Eskin and A. Okounkov, Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials,, Invent. Math., 145 (2001), 59. doi: 10.1007/s002220100142. Google Scholar [29] A. Eskin, A. Okounkov and R. Pandharipande, The theta characteristic of a branched covering,, Adv. Math., 217 (2008), 873. doi: 10.1016/j.aim.2006.08.001. Google Scholar [30] G. Forni, Deviations of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar [31] G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in Handbook of Dynamical Systems. Vol. 1B, (2006), 549. doi: 10.1016/S1874-575X(06)80033-0. Google Scholar [32] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle,, J. Mod. Dyn., 5 (2011), 355. doi: 10.3934/jmd.2011.5.355. Google Scholar [33] G. Forni, On the Brin prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations,, J. Mod. Dyn., 6 (2012), 139. doi: 10.3934/jmd.2012.6.139. Google Scholar [34] G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with degenerate Kontsevich-Zorich spectrum,, , (2008), 1. Google Scholar [35] G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar [36] G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle,, Ergodic Theory Dynam. Systems, 34 (2014), 353. doi: 10.1017/etds.2012.148. Google Scholar [37] G. Forni, C. Matheus and A. Zorich, Zero Lyapunov exponents of the Hodge bundle,, Comment. Math. Helv., 89 (2014), 489. doi: 10.4171/CMH/325. Google Scholar [38] R. Fox and R. Keshner, Concerning the transitive properties of geodesics on a rational polyhedron,, Duke Math. J., 2 (1936), 147. doi: 10.1215/S0012-7094-36-00213-2. Google Scholar [39] I. Gol'dsheĭd and G. Margulis, Lyapunov exponents of a product of random matrices (Russian),, Uspekhi Mat. Nauk, 44 (1989), 13. doi: 10.1070/RM1989v044n05ABEH002214. Google Scholar [40] Y. Guivarch and A. Raugi, Products of random matrices: Convergence theorems,, in Random Matrices and their Applications (Brunswick, (1984), 31. doi: 10.1090/conm/050/841080. Google Scholar [41] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar [42] F. Herrlich and G. Schmithüsen, An extraordinary origami curve,, Math. Nachr., 281 (2008), 219. doi: 10.1002/mana.200510597. Google Scholar [43] J. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1,, Matrix Editions, (2006). Google Scholar [44] P. Hubert and T. Schmidt, An introduction to Veech surfaces,, in Handbook of Dynamical Systems. Vol. 1B, (2006), 501. doi: 10.1016/S1874-575X(06)80031-7. Google Scholar [45] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Israel J. Math., 151 (2006), 281. doi: 10.1007/BF02777365. Google Scholar [46] E. 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##### References:
 [1] J. Athreya, Quantitative recurrence and large deviations for Teichmuller geodsic flow,, Geom. Dedicata, 119 (2006), 121. doi: 10.1007/s10711-006-9058-z. Google Scholar [2] J. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards,, Duke Math. J., 144 (2008), 285. doi: 10.1215/00127094-2008-037. Google Scholar [3] D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum,, , (2012). Google Scholar [4] A. Avila and G. Forni, Weak mixing for interval exchange transformations and translations flows,, Ann. of Math. (2), 165 (2007), 637. doi: 10.4007/annals.2007.165.637. Google Scholar [5] A. Avila and S. Gouëzel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow,, Ann. of Math. (2), 178 (2010), 385. doi: 10.4007/annals.2013.178.2.1. Google Scholar [6] A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow,, Pub. Math. Inst. Hautes Études Sci., 104 (2006), 143. doi: 10.1007/s10240-006-0001-5. Google Scholar [7] A. Avila, C. Matheus and J.-C. Yoccoz, On the Kontsevich-Zorich cocycle for McMullen's family of symmetric translation surfaces,, in preparation., (). Google Scholar [8] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1. Google Scholar [9] A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications,, Invent. Math., 181 (2010), 115. doi: 10.1007/s00222-010-0243-1. Google Scholar [10] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two,, Geom. Topol., 11 (2007), 1887. doi: 10.2140/gt.2007.11.1887. Google Scholar [11] N. Bergeron, Le Spectre des Surfaces Hyperboliques,, Savoirs Actuels (Les Ulis), (2011). Google Scholar [12] J. Borwein and P. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity,, Reprint of the 1987 original, (1987). Google Scholar [13] K. Burns, H. Masur and A. Wilkinson, The Weil-Petersson geodesic flow is ergodic,, Ann. of Math. (2), 175 (2012), 835. doi: 10.4007/annals.2012.175.2.8. Google Scholar [14] A. Bufetov, Limit theorems for translations flows,, Ann. of Math. (2), 179 (2014), 431. doi: 10.4007/annals.2014.179.2.2. Google Scholar [15] K. Calta, Veech surfaces and complete periodicity in genus two,, J. Amer. Math. Soc., 17 (2004), 871. doi: 10.1090/S0894-0347-04-00461-8. Google Scholar [16] D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying strata,, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 309. Google Scholar [17] Y. Cheung and A. Eskin, Unique ergodicity of translation flows,, in Partially Hyperbolic Dynamics, (2007), 213. Google Scholar [18] V. Delecroix, Cardinality of Rauzy classes,, Ann. Inst. Fourier (Grenoble), 63 (2013), 1651. doi: 10.5802/aif.2811. Google Scholar [19] P. Deligne, Un théorème de finitude pour la monodromie,, in Discrete Groups in Geometry and Analysis (New Haven, (1984), 1. doi: 10.1007/978-1-4899-6664-3_1. Google Scholar [20] V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, , (2011), 1. Google Scholar [21] V. Delecroix and C. Matheus, Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich,, , (2011), 1. Google Scholar [22] , Disquisitiones Mathematicae,, , (). Google Scholar [23] J. Ellenberg and D. B. McReynolds, Arithmetic Veech sublattices of $SL(2,\mathbbZ)$,, Duke Math. J., 161 (2012), 415. doi: 10.1215/00127094-1507412. Google Scholar [24] A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, Publ. Math. Inst. Hautes Études Sci., 120 (2014), 207. doi: 10.1007/s10240-013-0060-3. Google Scholar [25] A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319. Google Scholar [26] A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants,, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61. doi: 10.1007/s10240-003-0015-1. Google Scholar [27] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbbR)$ action on moduli space,, , (). Google Scholar [28] A. Eskin and A. Okounkov, Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials,, Invent. Math., 145 (2001), 59. doi: 10.1007/s002220100142. Google Scholar [29] A. Eskin, A. Okounkov and R. Pandharipande, The theta characteristic of a branched covering,, Adv. Math., 217 (2008), 873. doi: 10.1016/j.aim.2006.08.001. Google Scholar [30] G. Forni, Deviations of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar [31] G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in Handbook of Dynamical Systems. Vol. 1B, (2006), 549. doi: 10.1016/S1874-575X(06)80033-0. Google Scholar [32] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle,, J. Mod. Dyn., 5 (2011), 355. doi: 10.3934/jmd.2011.5.355. Google Scholar [33] G. Forni, On the Brin prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations,, J. Mod. Dyn., 6 (2012), 139. doi: 10.3934/jmd.2012.6.139. Google Scholar [34] G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with degenerate Kontsevich-Zorich spectrum,, , (2008), 1. Google Scholar [35] G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar [36] G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle,, Ergodic Theory Dynam. Systems, 34 (2014), 353. doi: 10.1017/etds.2012.148. Google Scholar [37] G. Forni, C. Matheus and A. Zorich, Zero Lyapunov exponents of the Hodge bundle,, Comment. Math. Helv., 89 (2014), 489. doi: 10.4171/CMH/325. Google Scholar [38] R. Fox and R. Keshner, Concerning the transitive properties of geodesics on a rational polyhedron,, Duke Math. J., 2 (1936), 147. doi: 10.1215/S0012-7094-36-00213-2. Google Scholar [39] I. Gol'dsheĭd and G. Margulis, Lyapunov exponents of a product of random matrices (Russian),, Uspekhi Mat. Nauk, 44 (1989), 13. doi: 10.1070/RM1989v044n05ABEH002214. Google Scholar [40] Y. Guivarch and A. Raugi, Products of random matrices: Convergence theorems,, in Random Matrices and their Applications (Brunswick, (1984), 31. doi: 10.1090/conm/050/841080. Google Scholar [41] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar [42] F. Herrlich and G. Schmithüsen, An extraordinary origami curve,, Math. Nachr., 281 (2008), 219. doi: 10.1002/mana.200510597. Google Scholar [43] J. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1,, Matrix Editions, (2006). Google Scholar [44] P. Hubert and T. Schmidt, An introduction to Veech surfaces,, in Handbook of Dynamical Systems. Vol. 1B, (2006), 501. doi: 10.1016/S1874-575X(06)80031-7. Google Scholar [45] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Israel J. Math., 151 (2006), 281. doi: 10.1007/BF02777365. Google Scholar [46] E. 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