# 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

 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. Google Scholar [2] J. Athreya, Quantitative recurrence and large deviations for the Teichmüller geodesic flow,, Geom. Dedicata, 119 (2006), 121. doi: 10.1007/s10711-006-9058-z. Google Scholar [3] A. Avila and A. Bufetov, Exponential decay of correlations for the Rauzy-Veech-Zorich induction map,, in, 51 (2007), 203. Google Scholar [4] A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation 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,, , (). Google Scholar [6] A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow,, Publ. Math. Inst. Hautes Études Sci., (2006), 143. doi: 10.1007/s10240-006-0001-5. Google Scholar [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., (). Google Scholar [8] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1. Google Scholar [9] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: A sufficient criterion,, Portugaliae Mathematica (N.S.), 64 (2007), 311. doi: 10.4171/PM/1789. Google Scholar [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. Google Scholar [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. Google Scholar [12] C. Bonatti and M. Viana, Lyapunov exponents with multiplicity $1$ for deterministic products of matrices,, Ergod. Th. Dynam. Sys., 24 (2004), 1295. doi: 10.1017/S0143385703000695. Google Scholar [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. doi: 10.1090/S0894-0347-06-00528-5. Google Scholar [14] A. Bufetov, Limit theorems for translation flows,, , (). Google Scholar [15] J. Chaika, There exists a topologically mixing IET,, , (). Google Scholar [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, (1982). Google Scholar [17] D. Dolgopyat, On decay of correlations in Anosov flows,, Ann. of Math. (2), 147 (1998), 357. Google Scholar [18] A. Eskin and G. A. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds,, in, (2004), 431. Google Scholar [19] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbb R)$ action on moduli space,, available from: , (). Google Scholar [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. doi: 10.2307/2952464. Google Scholar [21] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Annals of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar [22] G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, (2006), 549. Google Scholar [23] G. Forni, Sobolev regularity of solutions of the cohomological equation,, , (). Google Scholar [24] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle,, Journal of Modern Dynamics, 5 (2011), 355. doi: 10.3934/jmd.2011.5.355. Google Scholar [25] H. Furstenberg, "Stationary Processes and Prediction Theory,", Annals of Mathematics Studies, (1960). Google Scholar [26] H. Furstenberg, Noncommuting random products,, Transactions of the American Mathematical Society, 108 (1963), 377. doi: 10.1090/S0002-9947-1963-0163345-0. Google Scholar [27] H. Furstenberg and H. Kesten, Products of random matrices,, Ann. Math. Stat., 31 (1960), 457. doi: 10.1214/aoms/1177705909. Google Scholar [28] I. Y. Gold'sheuĭd and G. A. Margulis, Lyapunov indices of a product of random matrices,, Uspekhi Mat. Nauk, 44 (1989), 13. Google Scholar [29] Y. Guivarc'h and A. Raugi, Product of random matrices: Convergence theorems,, in, 50 (1989), 31. Google Scholar [30] A. B. Katok, Interval-exchange transformations and some special flows are not mixing,, Israel J. Math., 35 (1980), 301. doi: 10.1007/BF02760655. Google Scholar [31] A. B. Katok and A. M. Stepin, Approximations in ergodic theory,, Uspehi Mat. Nauk, 22 (1967), 81. doi: 10.1070/RM1967v022n05ABEH001227. Google Scholar [32] M. S. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25. doi: 10.1007/BF01236981. Google Scholar [33] M. S. Keane, Non-ergodic interval exchange transformations,, Israel J. Math., 26 (1977), 188. doi: 10.1007/BF03007668. Google Scholar [34] S. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations,, Erg. Theory and. Dynam. Sys., 5 (1985), 257. Google Scholar [35] M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318. Google Scholar [36] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. Google Scholar [37] F. Ledrappier, Positivity of the exponent for stationary sequences of matrices,, in, 1186 (1986), 56. Google Scholar [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. doi: 10.1090/S0894-0347-05-00490-X. Google Scholar [39] S. Marmi, P. Moussa and J.-C. Yoccoz, Linearization of generalized interval-exchange maps,, , (). Google Scholar [40] H. Masur, Extension of the Weyl-Petersson metric to the boundary of the Teichmüller space,, Duke Math. J., 43 (1976), 623. doi: 10.1215/S0012-7094-76-04350-7. Google Scholar [41] H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169. Google Scholar [42] H. Masur, Closed trajectories for quadratic differentials with an application to billiards,, Duke Math. J., 53 (1986), 307. doi: 10.1215/S0012-7094-86-05319-6. Google Scholar [43] A. Nogueira and D. Rudolph, Topological weak-mixing of interval-exchange maps,, Ergodic Theory Dynam. Systems, 17 (1997), 1183. doi: 10.1017/S0143385797086276. Google Scholar [44] V. I. Oseledec, The spectrum of ergodic automorphisms,, Dokl. Akad. Nauk SSSR, 168 (1966), 1009. Google Scholar [45] G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. Google Scholar [46] M. Ratner, The rate of mixing for geodesic and horocycle flows,, Ergodic Theory Dynam. Systems, 7 (1987), 267. Google Scholar [47] R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials,, to appear on Geom. Dedicata., (). Google Scholar [48] C. Ulcigrai, Absence of mixing in area-preserving flows on surfaces,, Annals of Mathematics (2), 173 (2011), 1743. doi: 10.4007/annals.2011.173.3.10. Google Scholar [49] W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations,, in, 10 (1981), 1979. Google Scholar [50] W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. Google Scholar [51] W. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties,, Amer. J. Math., 106 (1984), 1331. doi: 10.2307/2374396. Google Scholar [52] W. Veech, The Teichmüller geodesic flow,, Annals of Mathematics (2), 124 (1986), 441. doi: 10.2307/2007091. Google Scholar [53] M. Viana, Lyapunov exponents of Teichmüller flows,, in, 51 (2007), 139. Google Scholar [54] M. Viana, IMPA lectures on Lyapunov exponents, 2010., Available from: , (). Google Scholar [55] J.-C. Yoccoz, Interval-exchange maps and translation surfaces,, in, 10 (2010), 1. Google Scholar [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. Google Scholar [57] A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1994), 479. Google Scholar [58] A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents,, Ann. Inst. Fourier (Grenoble), 46 (1996), 325. doi: 10.5802/aif.1517. Google Scholar [59] A. Zorich, Deviation for interval exchange transformations,, Ergod. Th. & Dynam. Sys., 17 (1997), 1477. Google Scholar [60] A. Zorich, On hyperplane sections of periodic surfaces,, in, 179 (1997), 173. Google Scholar [61] A. Zorich, How do the leaves of a closed $1$-form wind around a surface?,, in, 197 (1999), 135. Google Scholar [62] A. Zorich, Flat surfaces,, in, (2006), 437. Google Scholar

show all references

##### References:
 [1] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Uspekhi Mat. Nauk, 18 (1963), 91. Google Scholar [2] J. Athreya, Quantitative recurrence and large deviations for the Teichmüller geodesic flow,, Geom. Dedicata, 119 (2006), 121. doi: 10.1007/s10711-006-9058-z. Google Scholar [3] A. Avila and A. Bufetov, Exponential decay of correlations for the Rauzy-Veech-Zorich induction map,, in, 51 (2007), 203. Google Scholar [4] A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation 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,, , (). Google Scholar [6] A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow,, Publ. Math. Inst. Hautes Études Sci., (2006), 143. doi: 10.1007/s10240-006-0001-5. Google Scholar [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., (). Google Scholar [8] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1. Google Scholar [9] A. Avila and M. Viana, Simplicity of Lyapunov Spectra: A sufficient criterion,, Portugaliae Mathematica (N.S.), 64 (2007), 311. doi: 10.4171/PM/1789. Google Scholar [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. Google Scholar [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. Google Scholar [12] C. Bonatti and M. Viana, Lyapunov exponents with multiplicity $1$ for deterministic products of matrices,, Ergod. Th. Dynam. Sys., 24 (2004), 1295. doi: 10.1017/S0143385703000695. Google Scholar [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. doi: 10.1090/S0894-0347-06-00528-5. Google Scholar [14] A. Bufetov, Limit theorems for translation flows,, , (). Google Scholar [15] J. Chaika, There exists a topologically mixing IET,, , (). Google Scholar [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, (1982). Google Scholar [17] D. Dolgopyat, On decay of correlations in Anosov flows,, Ann. of Math. (2), 147 (1998), 357. Google Scholar [18] A. Eskin and G. A. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds,, in, (2004), 431. Google Scholar [19] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbb R)$ action on moduli space,, available from: , (). Google Scholar [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. doi: 10.2307/2952464. Google Scholar [21] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Annals of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar [22] G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, (2006), 549. Google Scholar [23] G. Forni, Sobolev regularity of solutions of the cohomological equation,, , (). Google Scholar [24] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle,, Journal of Modern Dynamics, 5 (2011), 355. doi: 10.3934/jmd.2011.5.355. Google Scholar [25] H. Furstenberg, "Stationary Processes and Prediction Theory,", Annals of Mathematics Studies, (1960). Google Scholar [26] H. Furstenberg, Noncommuting random products,, Transactions of the American Mathematical Society, 108 (1963), 377. doi: 10.1090/S0002-9947-1963-0163345-0. Google Scholar [27] H. Furstenberg and H. Kesten, Products of random matrices,, Ann. Math. Stat., 31 (1960), 457. doi: 10.1214/aoms/1177705909. Google Scholar [28] I. Y. Gold'sheuĭd and G. A. Margulis, Lyapunov indices of a product of random matrices,, Uspekhi Mat. Nauk, 44 (1989), 13. Google Scholar [29] Y. Guivarc'h and A. Raugi, Product of random matrices: Convergence theorems,, in, 50 (1989), 31. Google Scholar [30] A. B. Katok, Interval-exchange transformations and some special flows are not mixing,, Israel J. Math., 35 (1980), 301. doi: 10.1007/BF02760655. Google Scholar [31] A. B. Katok and A. M. Stepin, Approximations in ergodic theory,, Uspehi Mat. Nauk, 22 (1967), 81. doi: 10.1070/RM1967v022n05ABEH001227. Google Scholar [32] M. S. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25. doi: 10.1007/BF01236981. Google Scholar [33] M. S. Keane, Non-ergodic interval exchange transformations,, Israel J. Math., 26 (1977), 188. doi: 10.1007/BF03007668. Google Scholar [34] S. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations,, Erg. Theory and. Dynam. Sys., 5 (1985), 257. Google Scholar [35] M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318. Google Scholar [36] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. Google Scholar [37] F. Ledrappier, Positivity of the exponent for stationary sequences of matrices,, in, 1186 (1986), 56. Google Scholar [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. doi: 10.1090/S0894-0347-05-00490-X. Google Scholar [39] S. Marmi, P. Moussa and J.-C. Yoccoz, Linearization of generalized interval-exchange maps,, , (). Google Scholar [40] H. Masur, Extension of the Weyl-Petersson metric to the boundary of the Teichmüller space,, Duke Math. J., 43 (1976), 623. doi: 10.1215/S0012-7094-76-04350-7. Google Scholar [41] H. Masur, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169. Google Scholar [42] H. Masur, Closed trajectories for quadratic differentials with an application to billiards,, Duke Math. J., 53 (1986), 307. doi: 10.1215/S0012-7094-86-05319-6. Google Scholar [43] A. Nogueira and D. Rudolph, Topological weak-mixing of interval-exchange maps,, Ergodic Theory Dynam. Systems, 17 (1997), 1183. doi: 10.1017/S0143385797086276. Google Scholar [44] V. I. Oseledec, The spectrum of ergodic automorphisms,, Dokl. Akad. Nauk SSSR, 168 (1966), 1009. Google Scholar [45] G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. Google Scholar [46] M. Ratner, The rate of mixing for geodesic and horocycle flows,, Ergodic Theory Dynam. Systems, 7 (1987), 267. Google Scholar [47] R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials,, to appear on Geom. Dedicata., (). Google Scholar [48] C. Ulcigrai, Absence of mixing in area-preserving flows on surfaces,, Annals of Mathematics (2), 173 (2011), 1743. doi: 10.4007/annals.2011.173.3.10. Google Scholar [49] W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations,, in, 10 (1981), 1979. Google Scholar [50] W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. Google Scholar [51] W. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties,, Amer. J. Math., 106 (1984), 1331. doi: 10.2307/2374396. Google Scholar [52] W. Veech, The Teichmüller geodesic flow,, Annals of Mathematics (2), 124 (1986), 441. doi: 10.2307/2007091. Google Scholar [53] M. Viana, Lyapunov exponents of Teichmüller flows,, in, 51 (2007), 139. Google Scholar [54] M. Viana, IMPA lectures on Lyapunov exponents, 2010., Available from: , (). Google Scholar [55] J.-C. Yoccoz, Interval-exchange maps and translation surfaces,, in, 10 (2010), 1. Google Scholar [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. Google Scholar [57] A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1994), 479. Google Scholar [58] A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents,, Ann. Inst. Fourier (Grenoble), 46 (1996), 325. doi: 10.5802/aif.1517. Google Scholar [59] A. Zorich, Deviation for interval exchange transformations,, Ergod. Th. & Dynam. Sys., 17 (1997), 1477. Google Scholar [60] A. Zorich, On hyperplane sections of periodic surfaces,, in, 179 (1997), 173. Google Scholar [61] A. Zorich, How do the leaves of a closed $1$-form wind around a surface?,, in, 197 (1999), 135. Google Scholar [62] A. Zorich, Flat surfaces,, in, (2006), 437. Google Scholar
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