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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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., No. 104 (2006), 143-211. 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), 1-56. 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), 311-376. 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), 865-874.  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), 579-624.  Google Scholar

[12]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity $1$ for deterministic products of matrices, Ergod. Th. Dynam. Sys., 24 (2004), 1295-1330. 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), 579-623. 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, New York, 1982.  Google Scholar

[17]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2), 147 (1998), 357-390.  Google Scholar

[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.  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), 295-344. 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), 1-103. doi: 10.2307/3062150.  Google Scholar

[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.  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), 355-395. doi: 10.3934/jmd.2011.5.355.  Google Scholar

[25]

H. Furstenberg, "Stationary Processes and Prediction Theory," Annals of Mathematics Studies, No. 44, Princeton University Press, Princeton, N.J., 1960.  Google Scholar

[26]

H. Furstenberg, Noncommuting random products, Transactions of the American Mathematical Society, 108 (1963), 377-428. 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), 457-469. 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), 13-60; translation in Russian Math. Surveys, 44 (1989), 11-71.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

M. S. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.  Google Scholar

[33]

M. S. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. 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), 257-271.  Google Scholar

[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.  Google Scholar

[36]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.  Google Scholar

[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.  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), 823-872. 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), 623-635. 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), 169-200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

[46]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  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), 1743-1778. doi: 10.4007/annals.2011.173.3.10.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

W. Veech, The Teichmüller geodesic flow, Annals of Mathematics (2), 124 (1986), 441-530. doi: 10.2307/2007091.  Google Scholar

[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.  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 "Homogeneous Flows, Moduli Spaces and Arithmetic," Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, (2010), 1-69.  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), 287-288. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[59]

A. Zorich, Deviation for interval exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 91-192; English translation: Russian Math. Surveys, 18 (1963), 85-191.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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., No. 104 (2006), 143-211. 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), 1-56. 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), 311-376. 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), 865-874.  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), 579-624.  Google Scholar

[12]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity $1$ for deterministic products of matrices, Ergod. Th. Dynam. Sys., 24 (2004), 1295-1330. 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), 579-623. 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, New York, 1982.  Google Scholar

[17]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2), 147 (1998), 357-390.  Google Scholar

[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.  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), 295-344. 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), 1-103. doi: 10.2307/3062150.  Google Scholar

[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.  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), 355-395. doi: 10.3934/jmd.2011.5.355.  Google Scholar

[25]

H. Furstenberg, "Stationary Processes and Prediction Theory," Annals of Mathematics Studies, No. 44, Princeton University Press, Princeton, N.J., 1960.  Google Scholar

[26]

H. Furstenberg, Noncommuting random products, Transactions of the American Mathematical Society, 108 (1963), 377-428. 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), 457-469. 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), 13-60; translation in Russian Math. Surveys, 44 (1989), 11-71.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

M. S. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.  Google Scholar

[33]

M. S. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. 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), 257-271.  Google Scholar

[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.  Google Scholar

[36]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.  Google Scholar

[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.  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), 823-872. 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), 623-635. 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), 169-200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

[46]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  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), 1743-1778. doi: 10.4007/annals.2011.173.3.10.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

W. Veech, The Teichmüller geodesic flow, Annals of Mathematics (2), 124 (1986), 441-530. doi: 10.2307/2007091.  Google Scholar

[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.  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 "Homogeneous Flows, Moduli Spaces and Arithmetic," Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, (2010), 1-69.  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), 287-288. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[59]

A. Zorich, Deviation for interval exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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The Editors. The 2009 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2010, 4 (2) : i-ii. doi: 10.3934/jmd.2010.4.2i

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The Editors. The 2015 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2016, 10: 173-174. doi: 10.3934/jmd.2016.10.173

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The Editors. The 2018 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2019, 15: 425-426. doi: 10.3934/jmd.2019025

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The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013

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The Editors. The 2017 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2019, 15: 131-132. doi: 10.3934/jmd.2019015

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Richard Tapia. My reflections on the Blackwell-Tapia prize. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1669-1672. doi: 10.3934/mbe.2013.10.1669

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Tao Zhang, W. Art Chaovalitwongse, Yue-Jie Zhang, P. M. Pardalos. The hot-rolling batch scheduling method based on the prize collecting vehicle routing problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 749-765. doi: 10.3934/jimo.2009.5.749

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Jian Sun, Haiyun Sheng, Yuefang Sun, Donglei Du, Xiaoyan Zhang. Approximation algorithm with constant ratio for stochastic prize-collecting Steiner tree problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021116

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