2019, 14: ⅴ-xxv. doi: 10.3934/jmd.2019v

Bill Veech's contributions to dynamical systems

1. 

Department of Mathematics, University of Maryland, College Park, MD 20742, USA

2. 

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA

3. 

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received  February 10, 2019 Published  February 2019

Citation: Giovanni Forni, Howard Masur, John Smillie. Bill Veech's contributions to dynamical systems. Journal of Modern Dynamics, 2019, 14: ⅴ-xxv. doi: 10.3934/jmd.2019v
References:
[1]

J. AuslanderG. Greschonig and A. Nagar, Reflections on equicontinuity, Proc. Amer. Math. Soc., 142 (2014), 3129-3137. doi: 10.1090/S0002-9939-2014-12034-X.

[2]

J. AthreyaA. BufetovA. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111. doi: 10.1215/00127094-1548443.

[3]

A. Avila and V. Delecroix, Weak mixing directions in non-arithmetic Veech surfaces, J. Amer. Math. Soc., 29 (2016), 1167-1208. doi: 10.1090/jams/856.

[4]

A. Avila and G. Forni, Weak mixing for interval exchange transformations, and translation flows, Ann. of Math., 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, Ann. of Math., 178 (2013), 385-442. doi: 10.4007/annals.2013.178.2.1.

[6]

A. AvilaS. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publications Mathématiques de l'IHÉS, 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5.

[7]

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.

[8]

M. Bainbridge, J. Smillie and B. Weiss, Horocycle dynamics: new invariants and eigenform loci in the stratum H(1,1), preprint, arXiv: 1603.00808.

[9]

D. Bernazzani, Most interval exchanges have no roots, J. Mod. Dyn., 11 (2017), 249-262. doi: 10.3934/jmd.2017011.

[10]

C. BoldrighiniM. Keane and F. Marchetti, Billiards in polygons, Ann. Probab., 6 (1978), 532-540. doi: 10.1214/aop/1176995475.

[11]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[12]

____, Rank two interval exchange transformations, Ergodic Theory and Dynamical Systems, 8(1988), 379–394. doi: 10.1017/S0143385700004521.

[13]

I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math., 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.

[14]

A. Bufetov, Logarithmic asymptotics for the number of periodic orbits of the Teichmüller flow on Veech's space of zippered rectangles, Mosc. Math. J., 9 (2009), 245-261. doi: 10.17323/1609-4514-2009-9-2-245-261.

[15]

K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8.

[16]

J. Chaika and A. Eskin, Self-Joinings for 3-IETs, preprint, arXiv: 1805.11167v2.

[17]

J. Chaika and R. Treviño, Logarithmic laws and unique ergodicity, J. Mod. Dyn., 11 (2017), 563-588. doi: 10.3934/jmd.2017022.

[18]

D. ChenM. Möller and D. Zagier, Quasimodularity and large genus limits of Siegel-Veech constants, J. Amer. Math. Soc., 31 (2018), 1059-1163. doi: 10.1090/jams/900.

[19]

A. Danilenko and A. Solomko, Simple mixing actions with uncountably many prime factors, Colloq. Math., 139 (2015), 37-54. doi: 10.4064/cm139-1-3.

[20]

D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems, Mosc. Math. J., 5 (2005), 55-67. doi: 10.17323/1609-4514-2005-5-1-55-66.

[21]

R. Ellis, The Veech structure theorem, Trans. of the Amer. Math. Soc., 186 (1973), 203-218. doi: 10.1090/S0002-9947-1973-0350712-1.

[22]

____, The Furstenberg structure theorem, Pacific Journal of Math., 76(1978), 345–349. doi: 10.2140/pjm.1978.76.345.

[23]

A. EskinM. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publications mathématiques de l'IHÉS, 120 (2014), 207-333. doi: 10.1007/s10240-013-0060-3.

[24]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Erg. Th. Dynam. Sys., 21 (2001), 443-478. doi: 10.1017/S0143385701001225.

[25]

A. EskinH. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants, Publications Mathématiques de l'IHÉS, 97 (2003), 61-179. doi: 10.1007/s10240-003-0015-1.

[26]

A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space, Journal of Modern Dynamics, 5 (2011), 71-105. doi: 10.3934/jmd.2011.5.71.

[27]

____, Invariant and stationary measures for the SL(2,ℝ) action on moduli space, Publications Mathématiques de l'IHÉS, 127 (2018), 95–324. doi: 10.1007/s10240-018-0099-2.

[28]

A. EskinM. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2,ℝ) action on moduli space, Ann. Math., 182 (2015), 673-721. doi: 10.4007/annals.2015.182.2.7.

[29]

A. EskinM. Mirzakhani and K. Rafi, Counting closed geodesics in strata, Invent. Math., 215 (2019), 535-607. doi: 10.1007/s00222-018-0832-y.

[30]

A. EskinG. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141. doi: 10.2307/120984.

[31]

S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle, Duke Math. J., 166 (2017), 657-706. doi: 10.1215/00127094-3715806.

[32]

G. Forni, Deviation of ergodic averages for area-preserving flows on higher genus surfaces, Ann. of Math., 155 (2002), 1-103. doi: 10.2307/3062150.

[33]

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____, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory and Dynamical Systems, 7 (1987), 149–153. doi: 10.1017/S0143385700003862.

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____, Decoding Rauzy induction: Bufetov’s question, Moscow Mathematical Journal, 10 (2010), 647–657. doi: 10.17323/1609-4514-2010-10-3-647-657.

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____, Möbius orthogonality for generalized Morse-Kakutani flows, American Journal of Mathematics, 139 (2017), 1157–1203. doi: 10.1353/ajm.2017.0031.

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____, Riemann sums and Möbius, Journal d'Analyse Mathématique, 135 (2018), 413–436. doi: 10.1007/s11854-018-0046-7.

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A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier, 46 (1996), 325-370. doi: 10.5802/aif.1517.

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____, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477–1499. doi: 10.1017/S0143385797086215.

show all references

References:
[1]

J. AuslanderG. Greschonig and A. Nagar, Reflections on equicontinuity, Proc. Amer. Math. Soc., 142 (2014), 3129-3137. doi: 10.1090/S0002-9939-2014-12034-X.

[2]

J. AthreyaA. BufetovA. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111. doi: 10.1215/00127094-1548443.

[3]

A. Avila and V. Delecroix, Weak mixing directions in non-arithmetic Veech surfaces, J. Amer. Math. Soc., 29 (2016), 1167-1208. doi: 10.1090/jams/856.

[4]

A. Avila and G. Forni, Weak mixing for interval exchange transformations, and translation flows, Ann. of Math., 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, Ann. of Math., 178 (2013), 385-442. doi: 10.4007/annals.2013.178.2.1.

[6]

A. AvilaS. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publications Mathématiques de l'IHÉS, 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5.

[7]

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.

[8]

M. Bainbridge, J. Smillie and B. Weiss, Horocycle dynamics: new invariants and eigenform loci in the stratum H(1,1), preprint, arXiv: 1603.00808.

[9]

D. Bernazzani, Most interval exchanges have no roots, J. Mod. Dyn., 11 (2017), 249-262. doi: 10.3934/jmd.2017011.

[10]

C. BoldrighiniM. Keane and F. Marchetti, Billiards in polygons, Ann. Probab., 6 (1978), 532-540. doi: 10.1214/aop/1176995475.

[11]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[12]

____, Rank two interval exchange transformations, Ergodic Theory and Dynamical Systems, 8(1988), 379–394. doi: 10.1017/S0143385700004521.

[13]

I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math., 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.

[14]

A. Bufetov, Logarithmic asymptotics for the number of periodic orbits of the Teichmüller flow on Veech's space of zippered rectangles, Mosc. Math. J., 9 (2009), 245-261. doi: 10.17323/1609-4514-2009-9-2-245-261.

[15]

K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8.

[16]

J. Chaika and A. Eskin, Self-Joinings for 3-IETs, preprint, arXiv: 1805.11167v2.

[17]

J. Chaika and R. Treviño, Logarithmic laws and unique ergodicity, J. Mod. Dyn., 11 (2017), 563-588. doi: 10.3934/jmd.2017022.

[18]

D. ChenM. Möller and D. Zagier, Quasimodularity and large genus limits of Siegel-Veech constants, J. Amer. Math. Soc., 31 (2018), 1059-1163. doi: 10.1090/jams/900.

[19]

A. Danilenko and A. Solomko, Simple mixing actions with uncountably many prime factors, Colloq. Math., 139 (2015), 37-54. doi: 10.4064/cm139-1-3.

[20]

D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems, Mosc. Math. J., 5 (2005), 55-67. doi: 10.17323/1609-4514-2005-5-1-55-66.

[21]

R. Ellis, The Veech structure theorem, Trans. of the Amer. Math. Soc., 186 (1973), 203-218. doi: 10.1090/S0002-9947-1973-0350712-1.

[22]

____, The Furstenberg structure theorem, Pacific Journal of Math., 76(1978), 345–349. doi: 10.2140/pjm.1978.76.345.

[23]

A. EskinM. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publications mathématiques de l'IHÉS, 120 (2014), 207-333. doi: 10.1007/s10240-013-0060-3.

[24]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Erg. Th. Dynam. Sys., 21 (2001), 443-478. doi: 10.1017/S0143385701001225.

[25]

A. EskinH. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants, Publications Mathématiques de l'IHÉS, 97 (2003), 61-179. doi: 10.1007/s10240-003-0015-1.

[26]

A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space, Journal of Modern Dynamics, 5 (2011), 71-105. doi: 10.3934/jmd.2011.5.71.

[27]

____, Invariant and stationary measures for the SL(2,ℝ) action on moduli space, Publications Mathématiques de l'IHÉS, 127 (2018), 95–324. doi: 10.1007/s10240-018-0099-2.

[28]

A. EskinM. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2,ℝ) action on moduli space, Ann. Math., 182 (2015), 673-721. doi: 10.4007/annals.2015.182.2.7.

[29]

A. EskinM. Mirzakhani and K. Rafi, Counting closed geodesics in strata, Invent. Math., 215 (2019), 535-607. doi: 10.1007/s00222-018-0832-y.

[30]

A. EskinG. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141. doi: 10.2307/120984.

[31]

S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle, Duke Math. J., 166 (2017), 657-706. doi: 10.1215/00127094-3715806.

[32]

G. Forni, Deviation of ergodic averages for area-preserving flows on higher genus surfaces, Ann. of Math., 155 (2002), 1-103. doi: 10.2307/3062150.

[33]

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Figure 1.  Rauzy diagram from Veech’s personal notes, June 21, 1977
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