American Institute of Mathematical Sciences

April  2012, 6(2): 139-182. doi: 10.3934/jmd.2012.6.139

On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations (Brin Prize article)

 1 Department of Mathematics, University of Maryland, College Park, MD 20742-4015

Published  August 2012

We review the Brin prize work of Artur Avila on Teichmüller dynamics and Interval Exchange Transformations. The paper is a nontechnical self-contained summary that intends to shed some light on Avila's early approach to the subject and on the significance of his achievements.
Citation: Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139
References:
 [1] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspekhi Mat. Nauk, 18 (1963), 91-192; English translation: Russian Math. Surveys, 18 (1963), 85-191. [2] J. Athreya, Quantitative recurrence and large deviations for the Teichmüller geodesic flow, Geom. Dedicata, 119 (2006), 121-140. doi: 10.1007/s10711-006-9058-z. [3] A. Avila and A. Bufetov, Exponential decay of correlations for the Rauzy-Veech-Zorich induction map, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," Fields Institute Communications, 51, Ameri. Math. Soc., Providence, RI, (2007), 203-211. [4] A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637. [5] A. Avila and S. Gouëzel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, arXiv:1011.5472. [6] A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., No. 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. [7] A. Avila and M. J. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials, to appear in Comm. Math. Helv. [8] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1. [9] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: A sufficient criterion, Portugaliae Mathematica (N.S.), 64 (2007), 311-376. doi: 10.4171/PM/1789. [10] V. Baladi and B. Vallée, Exponential decay of correlations for surface semi-flows without finite Markov partitions, Proc. Amer. Math. Soc., 133 (2005), 865-874. [11] C. Bonatti, X. Gómez-Mont and M. Viana, Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal. NonLinéaire, 20 (2003), 579-624. [12] C. Bonatti and M. Viana, Lyapunov exponents with multiplicity $1$ for deterministic products of matrices, Ergod. Th. Dynam. Sys., 24 (2004), 1295-1330. doi: 10.1017/S0143385703000695. [13] A. Bufetov, Decay of Correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials, J. Amer. Math. Soc., 19 (2006), 579-623. doi: 10.1090/S0894-0347-06-00528-5. [14] A. Bufetov, Limit theorems for translation flows, arXiv:0804.3970. [15] J. Chaika, There exists a topologically mixing IET, arXiv:0910.3986. [16] I. P. Cornfeld, S. V. Fomin and Y. G. Sinaui, "Ergodic Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982. [17] D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2), 147 (1998), 357-390. [18] A. Eskin and G. A. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, in "Random Walks and Geometry" (ed. V. A. Kaimanovich, in collaboration with K. Schmidt and W. Woess), Walter de Gruyter GmbH & Co. KG, Berlin, (2004), 431-444. [19] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbb R)$ action on moduli space, available from: http://math.uchicago.edu/~eskin/. [20] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464. [21] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Annals of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. [22] G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in "Handbook of Dynamical Systems," Vol. 1B (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, (2006), 549-580. [23] G. Forni, Sobolev regularity of solutions of the cohomological equation, arXiv:0707.0940. [24] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle, Journal of Modern Dynamics, 5 (2011), 355-395. doi: 10.3934/jmd.2011.5.355. [25] H. Furstenberg, "Stationary Processes and Prediction Theory," Annals of Mathematics Studies, No. 44, Princeton University Press, Princeton, N.J., 1960. [26] H. Furstenberg, Noncommuting random products, Transactions of the American Mathematical Society, 108 (1963), 377-428. doi: 10.1090/S0002-9947-1963-0163345-0. [27] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. [28] I. Y. Gold'sheuĭd and G. A. Margulis, Lyapunov indices of a product of random matrices, Uspekhi Mat. Nauk, 44 (1989), 13-60; translation in Russian Math. Surveys, 44 (1989), 11-71. [29] Y. Guivarc'h and A. Raugi, Product of random matrices: Convergence theorems, in "Random Matrices and their Applications" (Brunswick, Maine, 1984), Contemp. Math., 50, Amer. Math. Soc., Providence, RI, (1989), 31-54. [30] A. B. Katok, Interval-exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310. doi: 10.1007/BF02760655. [31] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk, 22 (1967), 81-106; translation in Russian Math. Survey, 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227. [32] M. S. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. [33] M. S. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668. [34] S. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Erg. Theory and. Dynam. Sys., 5 (1985), 257-271. [35] M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics'' (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Scientific Publ., River Edge, NJ, (1997), 318-332. [36] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. [37] F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, in "Lyapunov Exponents" (Bremen, 1984), Lect. Notes Math., 1186, Springer, Berlin, (1986), 56-73. [38] S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X. [39] S. Marmi, P. Moussa and J.-C. Yoccoz, Linearization of generalized interval-exchange maps, arXiv:1003.1191, to appear on Ann. of Math. [40] H. Masur, Extension of the Weyl-Petersson metric to the boundary of the Teichmüller space, Duke Math. J., 43 (1976), 623-635. doi: 10.1215/S0012-7094-76-04350-7. [41] H. Masur, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115, (1982), 169-200. [42] H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314. doi: 10.1215/S0012-7094-86-05319-6. [43] A. Nogueira and D. Rudolph, Topological weak-mixing of interval-exchange maps, Ergodic Theory Dynam. Systems, 17 (1997), 1183-1209. doi: 10.1017/S0143385797086276. [44] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR, 168 (1966), 1009-1011; translation in Soviet Math. Dokl., 7 (1966), 776-779. [45] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. [46] M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288. [47] R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, to appear on Geom. Dedicata. [48] C. Ulcigrai, Absence of mixing in area-preserving flows on surfaces, Annals of Mathematics (2), 173 (2011), 1743-1778. doi: 10.4007/annals.2011.173.3.10. [49] W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations, in "Ergodic Theory Dynam. Systems, I" (ed. A. Katok) (College Park, Md., 1979-80), Progr. Math., 10, Birkhäuser, Boston, Mass., (1981), 113-193. [50] W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391. [51] W. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math., 106 (1984), 1331-1359. doi: 10.2307/2374396. [52] W. Veech, The Teichmüller geodesic flow, Annals of Mathematics (2), 124 (1986), 441-530. doi: 10.2307/2007091. [53] M. Viana, Lyapunov exponents of Teichmüller flows, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," Fields Institute Communications, 51, Amer. Math. Soc., Providence, RI, (2007), 139-201. [54] M. Viana, IMPA lectures on Lyapunov exponents, 2010. Available from: http://w3.impa.br/ viana, (see w3.impa.br/ viana/out/lle.pdf). [55] J.-C. Yoccoz, Interval-exchange maps and translation surfaces, in "Homogeneous Flows, Moduli Spaces and Arithmetic," Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, (2010), 1-69. [56] A. Zorich, The S. P. Novikov problem on the semiclassical motion of an electron in homogeneous Magnetic Field, Russian Math. Surveys, 39 (1984), 287-288. [57] A. Zorich, Asymptotic flag of an orientable measured foliation on a surface, in "Geometric Study of Foliations'' (Tokyo, 1993), World Scientific Publ., River Edge, NJ, (1994), 479-498. [58] A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370. doi: 10.5802/aif.1517. [59] A. Zorich, Deviation for interval exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499. [60] A. Zorich, On hyperplane sections of periodic surfaces, in "Solitons, Geometry, and Topology: On the Crossroad" (eds. V. M. Buchstaber and S. P. Novikov), AMS Trasl. Ser. 2, 179, AMS, Providence, RI, (1997), 173-189. [61] A. Zorich, How do the leaves of a closed $1$-form wind around a surface?, in "Pseudoperiodic Topology'' (eds. Vladimir Arnold, Maxim Kontsevich and Anton Zorich), AMS Transl. Ser. 2, 197, AMS, Providence, RI, (1999), 135-178. [62] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics, and Geometry. 1" (eds. P. Cartier, B. Julia, P. Moussa and P. Vanhove), Springer, Berlin, (2006), 437-583.

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References:
 [1] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspekhi Mat. Nauk, 18 (1963), 91-192; English translation: Russian Math. Surveys, 18 (1963), 85-191. [2] J. Athreya, Quantitative recurrence and large deviations for the Teichmüller geodesic flow, Geom. Dedicata, 119 (2006), 121-140. doi: 10.1007/s10711-006-9058-z. [3] A. Avila and A. Bufetov, Exponential decay of correlations for the Rauzy-Veech-Zorich induction map, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," Fields Institute Communications, 51, Ameri. Math. Soc., Providence, RI, (2007), 203-211. [4] A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637. [5] A. Avila and S. Gouëzel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, arXiv:1011.5472. [6] A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., No. 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. [7] A. Avila and M. J. Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials, to appear in Comm. Math. Helv. [8] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1. [9] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: A sufficient criterion, Portugaliae Mathematica (N.S.), 64 (2007), 311-376. doi: 10.4171/PM/1789. [10] V. Baladi and B. Vallée, Exponential decay of correlations for surface semi-flows without finite Markov partitions, Proc. Amer. Math. Soc., 133 (2005), 865-874. [11] C. Bonatti, X. Gómez-Mont and M. Viana, Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal. NonLinéaire, 20 (2003), 579-624. [12] C. Bonatti and M. Viana, Lyapunov exponents with multiplicity $1$ for deterministic products of matrices, Ergod. Th. Dynam. Sys., 24 (2004), 1295-1330. doi: 10.1017/S0143385703000695. [13] A. Bufetov, Decay of Correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials, J. Amer. Math. Soc., 19 (2006), 579-623. doi: 10.1090/S0894-0347-06-00528-5. [14] A. Bufetov, Limit theorems for translation flows, arXiv:0804.3970. [15] J. Chaika, There exists a topologically mixing IET, arXiv:0910.3986. [16] I. P. Cornfeld, S. V. Fomin and Y. G. Sinaui, "Ergodic Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245, Springer-Verlag, New York, 1982. [17] D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2), 147 (1998), 357-390. [18] A. Eskin and G. A. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, in "Random Walks and Geometry" (ed. V. A. Kaimanovich, in collaboration with K. Schmidt and W. Woess), Walter de Gruyter GmbH & Co. KG, Berlin, (2004), 431-444. [19] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbb R)$ action on moduli space, available from: http://math.uchicago.edu/~eskin/. [20] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464. [21] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Annals of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. [22] G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in "Handbook of Dynamical Systems," Vol. 1B (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, (2006), 549-580. [23] G. Forni, Sobolev regularity of solutions of the cohomological equation, arXiv:0707.0940. [24] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle, Journal of Modern Dynamics, 5 (2011), 355-395. doi: 10.3934/jmd.2011.5.355. [25] H. Furstenberg, "Stationary Processes and Prediction Theory," Annals of Mathematics Studies, No. 44, Princeton University Press, Princeton, N.J., 1960. [26] H. Furstenberg, Noncommuting random products, Transactions of the American Mathematical Society, 108 (1963), 377-428. doi: 10.1090/S0002-9947-1963-0163345-0. [27] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. [28] I. Y. Gold'sheuĭd and G. A. Margulis, Lyapunov indices of a product of random matrices, Uspekhi Mat. Nauk, 44 (1989), 13-60; translation in Russian Math. Surveys, 44 (1989), 11-71. [29] Y. Guivarc'h and A. Raugi, Product of random matrices: Convergence theorems, in "Random Matrices and their Applications" (Brunswick, Maine, 1984), Contemp. Math., 50, Amer. Math. Soc., Providence, RI, (1989), 31-54. [30] A. B. Katok, Interval-exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310. doi: 10.1007/BF02760655. [31] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk, 22 (1967), 81-106; translation in Russian Math. Survey, 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227. [32] M. S. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. [33] M. S. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668. [34] S. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Erg. Theory and. Dynam. Sys., 5 (1985), 257-271. [35] M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics'' (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Scientific Publ., River Edge, NJ, (1997), 318-332. [36] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. [37] F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, in "Lyapunov Exponents" (Bremen, 1984), Lect. Notes Math., 1186, Springer, Berlin, (1986), 56-73. [38] S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X. [39] S. Marmi, P. Moussa and J.-C. Yoccoz, Linearization of generalized interval-exchange maps, arXiv:1003.1191, to appear on Ann. of Math. [40] H. Masur, Extension of the Weyl-Petersson metric to the boundary of the Teichmüller space, Duke Math. J., 43 (1976), 623-635. doi: 10.1215/S0012-7094-76-04350-7. [41] H. Masur, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115, (1982), 169-200. [42] H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314. doi: 10.1215/S0012-7094-86-05319-6. [43] A. Nogueira and D. Rudolph, Topological weak-mixing of interval-exchange maps, Ergodic Theory Dynam. Systems, 17 (1997), 1183-1209. doi: 10.1017/S0143385797086276. [44] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR, 168 (1966), 1009-1011; translation in Soviet Math. Dokl., 7 (1966), 776-779. [45] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. [46] M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288. [47] R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, to appear on Geom. Dedicata. [48] C. Ulcigrai, Absence of mixing in area-preserving flows on surfaces, Annals of Mathematics (2), 173 (2011), 1743-1778. doi: 10.4007/annals.2011.173.3.10. [49] W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations, in "Ergodic Theory Dynam. Systems, I" (ed. A. Katok) (College Park, Md., 1979-80), Progr. Math., 10, Birkhäuser, Boston, Mass., (1981), 113-193. [50] W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391. [51] W. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math., 106 (1984), 1331-1359. doi: 10.2307/2374396. [52] W. Veech, The Teichmüller geodesic flow, Annals of Mathematics (2), 124 (1986), 441-530. doi: 10.2307/2007091. [53] M. Viana, Lyapunov exponents of Teichmüller flows, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," Fields Institute Communications, 51, Amer. Math. Soc., Providence, RI, (2007), 139-201. [54] M. Viana, IMPA lectures on Lyapunov exponents, 2010. Available from: http://w3.impa.br/ viana, (see w3.impa.br/ viana/out/lle.pdf). [55] J.-C. Yoccoz, Interval-exchange maps and translation surfaces, in "Homogeneous Flows, Moduli Spaces and Arithmetic," Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, (2010), 1-69. [56] A. Zorich, The S. P. Novikov problem on the semiclassical motion of an electron in homogeneous Magnetic Field, Russian Math. Surveys, 39 (1984), 287-288. [57] A. Zorich, Asymptotic flag of an orientable measured foliation on a surface, in "Geometric Study of Foliations'' (Tokyo, 1993), World Scientific Publ., River Edge, NJ, (1994), 479-498. [58] A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370. doi: 10.5802/aif.1517. [59] A. Zorich, Deviation for interval exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499. [60] A. Zorich, On hyperplane sections of periodic surfaces, in "Solitons, Geometry, and Topology: On the Crossroad" (eds. V. M. Buchstaber and S. P. Novikov), AMS Trasl. Ser. 2, 179, AMS, Providence, RI, (1997), 173-189. [61] A. Zorich, How do the leaves of a closed $1$-form wind around a surface?, in "Pseudoperiodic Topology'' (eds. Vladimir Arnold, Maxim Kontsevich and Anton Zorich), AMS Transl. Ser. 2, 197, AMS, Providence, RI, (1999), 135-178. [62] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics, and Geometry. 1" (eds. P. Cartier, B. Julia, P. Moussa and P. Vanhove), Springer, Berlin, (2006), 437-583.
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