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Bill Veech's contributions to dynamical systems

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

  • [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.
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    [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.
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    [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] ____, On the Lyapunov exponents of the Kontsevich–Zorich cocycle, in Handbook of Dynamical Systems, 1B (eds. B. Hasselblatt and A. Katok), Elsevier, 2006,549–580.
    [34] G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich–Zorich spectrum, preprint, 2008, arXiv: 0810.0023.
    [35] ____, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dynam., 8 (2014), 271–436. doi: 10.3934/jmd.2014.8.271.
    [36] G. ForniC. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.  doi: 10.3934/jmd.2011.5.285.
    [37] ____, Zero Lyapunov exponents of the Hodge bundle, Comment. Math. Helv., 89 (2014), 489–535. doi: 10.4171/CMH/325.
    [38] H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.
    [39] E. Glasner and B. Weiss, A simple weakly mixing transformation with nonunique prime factors, Amer. J. Math., 116 (1994), 361-375.  doi: 10.2307/2374933.
    [40] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213.  doi: 10.1215/S0012-7094-00-10321-3.
    [41] F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Mathematische Nachrichten, 281 (2008), 219-237.  doi: 10.1002/mana.200510597.
    [42] P. Hubert and T. A. Schmidt, Infinitely generated Veech groups, Duke Math. J., 123 (2004), 49-69.  doi: 10.1215/S0012-7094-04-12312-8.
    [43] A. del Junco, A simple map with no prime factors, Israel J. Math., 104 (1998), 301–320. doi: 10.1007/BF02897068.
    [44] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems, 7 (1987), 531-557.  doi: 10.1017/S0143385700004193.
    [45] ____,A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems, 7 (1987), 229–247. doi: 10.1017/S0143385700003977.
    [46] A. B. Katok, Invariant measures of flows on oriented surfaces, Soviet Math. Dokl., 14 (1973), 1104-1108. 
    [47] ____, Interval exchange transformations and some special flows are not mixing, Israel Journal of Mathematics, 35 (1980), 301–310. doi: 10.1007/BF02760655.
    [48] A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Math. Res. Lett., 3 (1996), 191-210.  doi: 10.4310/MRL.1996.v3.n2.a6.
    [49] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk, 22 (1967), 81-106. 
    [50] A. B. Katok and A. M. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes of the Academy of Sciences of the USSR, 18 (1975), 760–764; errata, 20 (1976), 1051.
    [51] M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.  doi: 10.1007/BF01236981.
    [52] ____, Non-ergodic interval exchange transformations, Israel Journal of Mathematics, 26 (1977), 188–196. doi: 10.1007/BF03007668.
    [53] R. Kenyon and J. Smillie, Billiards in rational-angled triangles, Comment. Mathem. Helv., 75 (2000), 65-108.  doi: 10.1007/s000140050113.
    [54] S. P. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergodic Theory and Dynamical Systems, 5 (1985), 257-271.  doi: 10.1017/S0143385700002881.
    [55] S. P. KerckhoffH. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Mathematics, 124 (1986), 293-311.  doi: 10.2307/1971280.
    [56] H. B. Keynes and D. Newton, A 'minimal', non-uniquely ergodic interval exchange transformation, Mathematische Zeitschrift, 148 (1976), 101-105.  doi: 10.1007/BF01214699.
    [57] M. Kontsevich, Lyapunov exponents and Hodge theory, in The Mathematical Beauty of Physics: A Memorial Volume for Claude Itzykson (eds. J. M. Drouffe and J. B. Zuber), Saclay, France 5-7 June 1996, Advanced Series in Mathematical Physics, 24, World Scientific Pub. Co. Inc., River Edge, NJ, 1997.
    [58] M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory, preprint, 1997, arXiv: hep-th/9701164v1.
    [59] J. Marklof and A. Strömbergsson, Free Path Lengths in Quasicrystals, Communications in Mathematical Physics, 330 (2014), 723-755.  doi: 10.1007/s00220-014-2011-3.
    [60] H. Masur, Interval exchange transformations and measured foliations, Annals of Mathematics, 115 (1982), 169-200.  doi: 10.2307/1971341.
    [61] ____, Ergodic actions of the mapping class group, Proc. A.M.S., 94 (1985), 455–459. doi: 10.1090/S0002-9939-1985-0787893-5.
    [62] ____, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in Holomorphic Functions and Moduli (eds. D. Drasin, I. Kra, C. J. Earle, A. Marden and F. W. Gehring), Mathematical Sciences Research Institute Publications, 10, Springer, New York, NY, 1988,215–228. doi: 10.1007/978-1-4613-9602-4_20.
    [63] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885.  doi: 10.1090/S0894-0347-03-00432-6.
    [64] ____, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191–223. doi: 10.1007/BF02392964.
    [65] J.-Ch. Puchta, On triangular billiards, Comment. Mathem. Helv., 76 (2001), 501-505.  doi: 10.1007/PL00013215.
    [66] G. Rauzy, Échanges d' intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  doi: 10.4064/aa-34-4-315-328.
    [67] M. Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergodic Theory and Dynamical Systems, 1 (1981), 461-488.  doi: 10.1017/s0143385700001383.
    [68] A. Sauvaget, Volumes and Siegel-Veech constants of $\mathcal H(2g-2)$ and Hodge integrals, Geometric and Functional Analysis, 28 (2018), 1756-1779.  doi: 10.1007/s00039-018-0468-5.
    [69] J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260.  doi: 10.1007/BF02771535.
    [70] R. Treviño, On the ergodicity of flat surfaces of finite area, Geometric and Functional Analysis, 24 (2014), 360-386.  doi: 10.1007/s00039-014-0269-4.
    [71] W. A. Veech, Almost automorphic functions on groups, American Journal of Mathematics, 87 (1965), 719-751.  doi: 10.2307/2373071.
    [72] ____, The equicontinuous structure relation for minimal Abelian transformation groups, American Journal of Mathematics, 90 (1968), 723–732. doi: 10.2307/2373480.
    [73] ____, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2, Transactions of the American Mathematical Society, 140 (1969), 1–33. doi: 10.2307/1995120.
    [74] ____, Point-distal flows, American Journal of Mathematics, 92 (1970), 205–242. doi: 10.2307/2373504.
    [75] ____, Topological Dynamics, Bulletin of American Mathematical Society, 83 (1977), 775–830. doi: 10.1090/S0002-9904-1977-14319-X.
    [76] ____, Interval exchange transformations, Journal d'Analyse Mathématique, 33 (1978), 222–272. doi: 10.1007/BF02790174.
    [77] ____, Projective Swiss cheeses and uniquely ergodic interval exchange transformations, in Ergodic Theory and Dynamical Systems (ed. A. Katok), Progress in Mathematics, 10, Birkhäuser, Boston, MA, 1981,113–193. doi: 10.1007/978-1-4899-6696-4_5.
    [78] ____, A Gauss measure on the set of interval exchange transformations, Proceedings of the National Academy of Sciences of the United States of America, 78 (1981), 696–697. doi: 10.1073/pnas.78.2.696.
    [79] ____, Gauss measures for transformations on the space of interval exchange maps, Annals of Mathematics, 115 (1982), 201–242. doi: 10.2307/1971391.
    [80] ____, A criterion for a process to be prime, Monatshefte für Mathematik, 94 (1982), 335–341. doi: 10.1007/BF01667386.
    [81] ____, he metric theory of interval exchange transformations Ⅰ. Generic spectral properties, American Journal of Mathematics, 106 (1984), 1331–1359. doi: 10.2307/2374396.
    [82] ____, The metric theory of interval exchange transformations Ⅱ. Approximation by primitive interval exchanges, American Journal of Mathematics, 106 (1984), 1361–1387. doi: 10.2307/2374397.
    [83] ____, The metric theory of interval exchange transformations Ⅲ. The Sah-Arnoux-Fathi invariant, American Journal of Mathematics, 106 (1984), 1389–1422. doi: 10.2307/2374398.
    [84] ____, Dynamics over Teichmüller space, Bulletin of the American Mathematical Society, 14 (1986), 103–106. doi: 10.1090/S0273-0979-1986-15406-6.
    [85] ____, The Teichmüller geodesic flow, Annals of Mathematics, 124 (1986), 441–530. doi: 10.2307/2007091.
    [86] ____, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynamical Systems, 6 (1986), 449–473. doi: 10.1017/S0143385700003606.
    [87] ____, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory and Dynamical Systems, 7 (1987), 149–153. doi: 10.1017/S0143385700003862.
    [88] ____, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae, 97 (1989), 553–583. doi: 10.1007/BF01388890.
    [89] ____, The billiard in a regular polygon, Geometric and Functional Analysis GAFA, 2 (1992), 341–379. doi: 10.1007/BF01896876.
    [90] ____, Geometric realizations of hyperelliptic curves, in Algorithms, Fractals, and Dynamics (ed. Y. Takahashi), Springer, Boston, MA, 1995,217–226. doi: 10.1007/978-1-4613-0321-3_19.
    [91] ____, Siegel measures, Annals of Mathematics, 148 (1998), 895–944. doi: 10.2307/121033.
    [92] ____, Decoding Rauzy induction: Bufetov’s question, Moscow Mathematical Journal, 10 (2010), 647–657. doi: 10.17323/1609-4514-2010-10-3-647-657.
    [93] ____, Invariant distributions for interval exchange transformations, in Dynamical Systems and Group Actions (eds. L. Bowen, R. Grigorchuk and Y. Vorobets), Contemporary Mathematics, 567, American Mathematical Soc., 2012,191–220. doi: 10.1090/conm/567.
    [94] ____, Möbius orthogonality for generalized Morse-Kakutani flows, American Journal of Mathematics, 139 (2017), 1157–1203. doi: 10.1353/ajm.2017.0031.
    [95] ____, Riemann sums and Möbius, Journal d'Analyse Mathématique, 135 (2018), 413–436. doi: 10.1007/s11854-018-0046-7.
    [96] Y. B. Vorobets, Planar structures and billiards in rational polygons: The Veech alternative, Russian Mathematical Surveys, 51 (1996), 779-817.  doi: 10.1070/RM1996v051n05ABEH002993.
    [97] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165. 
    [98] 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.
    [99] ____, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477–1499. doi: 10.1017/S0143385797086215.
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