American Institute of Mathematical Sciences

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

J. Athreya, A. Bufetov, A. 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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

A. Avila, S. 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.  Google Scholar

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

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

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

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

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.  Google Scholar

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

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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

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

D. Chen, M. 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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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

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

A. Eskin, M. 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.  Google Scholar

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

A. Eskin, H. 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.  Google Scholar

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.  Google Scholar

____, 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.  Google Scholar

A. Eskin, M. 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.  Google Scholar

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

A. Eskin, G. 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.  Google Scholar

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

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.  Google Scholar

____, 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. Google Scholar

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich–Zorich spectrum, preprint, 2008, arXiv: 0810.0023. Google Scholar

____, 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.  Google Scholar

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.  doi: 10.3934/jmd.2011.5.285.  Google Scholar

____, Zero Lyapunov exponents of the Hodge bundle, Comment. Math. Helv., 89 (2014), 489–535. doi: 10.4171/CMH/325.  Google Scholar

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.  Google Scholar

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.  Google Scholar

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.  Google Scholar

F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Mathematische Nachrichten, 281 (2008), 219-237.  doi: 10.1002/mana.200510597.  Google Scholar

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.  Google Scholar

A. del Junco, A simple map with no prime factors, Israel J. Math., 104 (1998), 301–320. doi: 10.1007/BF02897068.  Google Scholar

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.  Google Scholar

____,A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems, 7 (1987), 229–247. doi: 10.1017/S0143385700003977.  Google Scholar

A. B. Katok, Invariant measures of flows on oriented surfaces, Soviet Math. Dokl., 14 (1973), 1104-1108.   Google Scholar

____, Interval exchange transformations and some special flows are not mixing, Israel Journal of Mathematics, 35 (1980), 301–310. doi: 10.1007/BF02760655.  Google Scholar

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.  Google Scholar

A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk, 22 (1967), 81-106.   Google Scholar

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. Google Scholar

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

____, Non-ergodic interval exchange transformations, Israel Journal of Mathematics, 26 (1977), 188–196. doi: 10.1007/BF03007668.  Google Scholar

R. Kenyon and J. Smillie, Billiards in rational-angled triangles, Comment. Mathem. Helv., 75 (2000), 65-108.  doi: 10.1007/s000140050113.  Google Scholar

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.  Google Scholar

S. P. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Mathematics, 124 (1986), 293-311.  doi: 10.2307/1971280.  Google Scholar

H. B. Keynes and D. Newton, A 'minimal', non-uniquely ergodic interval exchange transformation, Mathematische Zeitschrift, 148 (1976), 101-105.  doi: 10.1007/BF01214699.  Google Scholar

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.  Google Scholar

M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory, preprint, 1997, arXiv: hep-th/9701164v1. Google Scholar

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.  Google Scholar

H. Masur, Interval exchange transformations and measured foliations, Annals of Mathematics, 115 (1982), 169-200.  doi: 10.2307/1971341.  Google Scholar

____, Ergodic actions of the mapping class group, Proc. A.M.S., 94 (1985), 455–459. doi: 10.1090/S0002-9939-1985-0787893-5.  Google Scholar

____, 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.  Google Scholar

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.  Google Scholar

____, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191–223. doi: 10.1007/BF02392964.  Google Scholar

J.-Ch. Puchta, On triangular billiards, Comment. Mathem. Helv., 76 (2001), 501-505.  doi: 10.1007/PL00013215.  Google Scholar

G. Rauzy, Échanges d' intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  doi: 10.4064/aa-34-4-315-328.  Google Scholar

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.  Google Scholar

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.  Google Scholar

J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260.  doi: 10.1007/BF02771535.  Google Scholar

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.  Google Scholar

W. A. Veech, Almost automorphic functions on groups, American Journal of Mathematics, 87 (1965), 719-751.  doi: 10.2307/2373071.  Google Scholar

____, The equicontinuous structure relation for minimal Abelian transformation groups, American Journal of Mathematics, 90 (1968), 723–732. doi: 10.2307/2373480.  Google Scholar

____, 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.  Google Scholar

____, Point-distal flows, American Journal of Mathematics, 92 (1970), 205–242. doi: 10.2307/2373504.  Google Scholar

____, Topological Dynamics, Bulletin of American Mathematical Society, 83 (1977), 775–830. doi: 10.1090/S0002-9904-1977-14319-X.  Google Scholar

____, Interval exchange transformations, Journal d'Analyse Mathématique, 33 (1978), 222–272. doi: 10.1007/BF02790174.  Google Scholar

____, 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.  Google Scholar

____, 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.  Google Scholar

____, Gauss measures for transformations on the space of interval exchange maps, Annals of Mathematics, 115 (1982), 201–242. doi: 10.2307/1971391.  Google Scholar

____, A criterion for a process to be prime, Monatshefte für Mathematik, 94 (1982), 335–341. doi: 10.1007/BF01667386.  Google Scholar

____, he metric theory of interval exchange transformations Ⅰ. Generic spectral properties, American Journal of Mathematics, 106 (1984), 1331–1359. doi: 10.2307/2374396.  Google Scholar

____, 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.  Google Scholar

____, 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.  Google Scholar

____, Dynamics over Teichmüller space, Bulletin of the American Mathematical Society, 14 (1986), 103–106. doi: 10.1090/S0273-0979-1986-15406-6.  Google Scholar

____, The Teichmüller geodesic flow, Annals of Mathematics, 124 (1986), 441–530. doi: 10.2307/2007091.  Google Scholar

____, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynamical Systems, 6 (1986), 449–473. doi: 10.1017/S0143385700003606.  Google Scholar

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

____, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae, 97 (1989), 553–583. doi: 10.1007/BF01388890.  Google Scholar

____, The billiard in a regular polygon, Geometric and Functional Analysis GAFA, 2 (1992), 341–379. doi: 10.1007/BF01896876.  Google Scholar

____, 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.  Google Scholar

____, Siegel measures, Annals of Mathematics, 148 (1998), 895–944. doi: 10.2307/121033.  Google Scholar

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

____, 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.  Google Scholar

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

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

Y. B. Vorobets, Planar structures and billiards in rational polygons: The Veech alternative, Russian Mathematical Surveys, 51 (1996), 779-817.  doi: 10.1070/RM1996v051n05ABEH002993.  Google Scholar

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165.   Google Scholar

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.  Google Scholar

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