Abstract
We introduce Lagrange Spectra of closed-invariant loci for the action of SL(2, \({\mathbb{R}}\)) on the moduli space of translation surfaces, generalizing the classical Lagrange Spectrum, and we analyze them with renormalization techniques. A formula for the values in such spectra is established in terms of the Rauzy–Veech induction and it is used to show that any invariant locus has closed Lagrange spectrum and values corresponding to pseudo-Anosov elements are dense. Moreover we show that Lagrange spectra of arithmetic Teichmüller discs contain an Hall’s ray, giving an explicit bound for it via a second formula which uses the classical continued fraction algorithm. In addition, we show the equivalence of several definitions of bounded Teichmüller geodesics and bounded type interval exchange transformations and we prove quantitative estimates on excursions to the boundary of moduli space in terms of norms of positive matrices in the Rauzy–Veech induction.
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Avila A., Gouezel S., Yoccoz J.-C.: Exponential mixing for the Teichmüller flow. Publications mathématiques de l’IHES, 104, 143–211 (2006)
M. Artigiani, L. Marchese and C. Ulcigrai. The Lagrange Spectrum of a Veech surface has a Hall ray (2014). arXiv:1409.7023.
Birkhoff G.: Extensions of Jentzsth’s theorem. Transactions of the American Mathematical Society, 85, 219–227 (1957)
Boshernitzan M.: A condition for minimal interval exchange maps to be uniquely ergodic.. Duke Mathematical Journal, 52, 723–752 (1985)
T.W. Cusick and M.E. Flahive. The Markoff and Lagrange spectra. Mathematicas Surveys and Monographs, Vol. 30 (1989).
A. Eskin and M. Mirzakhani. Invariant and stationary measures for the \({\rm SL(2,\mathbb{R}})\) action on moduli space. (2013). arXiv:1302.3320.
A. Eskin, K. Rafi, and M. Mirzakhani. Counting closed geodesics in strata. (2012). arXiv:1206.5574.
S. Ferenczi. Dynamical Generalizations of the Lagrange Spectrum (2010). arXiv:1108.3628.
Hall M.: On the sum and products of continued fractions. Annals of Mathematics, 48, 966–993 (1947)
Haas A., Series C.: The Hurwitz constant and Diophantine approximation on Hecke groups. Journal of London Mathematical Society, 2(34), 219–334 (1986)
Hamenstädt U.: Dynamics of the Teichmueller flow on compact invariant sets. Journal of Modern Dynamics, 4, 393–418 (2010)
Hersonsky S., Paulin F.: On the almost sure spiraling of geodesics in negatively curved manifolds. Journal of Differential Geometry, 2(85), 271–314 (2010)
Hubert P., Lelièvre S.: Prime arithmetic Teichmüller disc in \({\mathcal{H}(2)}\). Israel Journal of Mathematics, 151, 281–321 (2006)
D.H. Kim and S. Marmi. Bounded type interval exchange maps (2013). arXiv:1307.3511.
A.Y. Khinchin. Continued fractions. P. Noordhoff, Groningen (1963) (English translation).
Kleinboch D., Weiss B.: Bounded geodesics in moduli space. International Mathematical Research Notices, 30, 1551–1560 (2004)
Marchese L.: The Khinchin theorem for interval exchange transformations. Journal of Modern Dynamics, 1(5), 123–183 (2011)
Marmi S., Moussa P., Yoccoz J.-C.: The cohomological equation for Roth-type interval exchange maps. Journal of American Mathematical Society, 4(18), 823–872 (2005)
Maucourant F.: “Sur les spectres de Lagrange et de Markoff des corps imaginaires quadratiques” (French). Ergodic Theory and Dynamical Systems, 1(23), 193–205 (2003)
Masur H.: Interval exchange transformation and measured foliations. Annals of Mathematics, 115, 169–200 (1982)
G. Moreira.“Introduçao à à teoria dos números” (Portuguese) [Introduction to number theory] Monografías del Instituto de Matemática y Ciencias Afines, Vol. 24. Instituto de Matemática y Ciencias Afines, IMCA, Lima; Pontificia Universidad Católica del Perú, Lima (2002). ISBN: 9972-899-01-2,11-01
G. Moreira and S. Romaña. On the Lagrange and Markov dynamical spectra (2013). arXiv:1310.3903.
J. Parkkonen and F. Paulin. On the closedness of approximation spectra. Journal de Théorie des Nombres de Bordeaux, (3)21 (2009), 701–710.
Parkkonen J., Paulin F.: Prescribing the behaviour of geodesics in negative curvature. Geometry and Topology, 1(14), 277–392 (2010)
Parkkonen J., Paulin F.: Spiraling spectra of geodesic lines in negatively curved manifolds. Mathematische Zeitschrift, 1(−2268), 101–142 (2011)
Rauzy G.: Echanges d’intervalles et transformations induites (French). Acta Arithmetica, 4(34), 315–328 (1979)
Schmidt T.A., Sheingorn M.: Riemann surfaces have Hall rays at each cusp. Illinois Journal of Mathematics, 3(41), 378–397 (1997)
Series C.: The geometry of Markoff numbers. The Mathematical Intelligencer, 7, 20–29 (1985)
Series C.: The modular surface and continued fractions. Journal of the London Mathematical Society, 31(69–80), 31 69–80 (1985)
C. Series. Geometrical methods of symbolic coding. In: T. Bedford, M. Keane, and C. Series (eds.) Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces. Oxford University Press, New York (1991), pp. 121–151.
Smillie J., Weiss B.: Characterizations of lattice surfaces. Inventiones Mathematicae, 3(180), 535–557 (2010)
Veech W.: Gauss measures for transformations on the space of interval exchange maps. Annals of Mathematics, 115, 201–242 (1982)
W. Veech. Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation. Ergodic Theory and Dynamical Systems, (1)7 (1987), 27–48
W. Veech. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Inventiones Mathematicae, (3)97 (1989), 553–583.
Y. Vorobets. Planar structures and billiards in rational polygons: the Veech alternative. (Russian).Uspekhi Mat. Nauk, 51(5) (1996), 3–42 (translation in Russian Mathematical Surveys, (5)51 (1996), 779–817).
Vulakh L.: The Markov spectra for triangle groups. Journal of Number Theory, 1(67), 11–28 (1997)
Vulakh L.: The Markov spectra for Fuchsian groups. Transactions of the American Mathematical Society, 352, 4067–4094 (2000)
Vulakh L.: Diophantine approximation in R n. Transactions of the American Mathematical Society, 2(347), 573–585 (1995)
Vulakh L.: Farey polytopes and continued fractions associated with discrete hyperbolic groups. Transactions of the American Mathematical Society, 351, 2295–2323 (1999)
Vulakh L.: Diophantine approximation on Bianchi groups. Journal of Number Theory, 1(54), 73–80 (1995)
A. Wright. The field of definition of affine invariant submanifolds of the moduli space of abelian differentials (2012). arXiv:1210.4806.
J.-C. Yoccoz. Echanges d’intervalles. Cours Collège de France (2005).
Zorich A.: Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Annales de l’Institut Fourier, 2(46), 325–370 (1996)
A. Zorich. Flat surfaces. In: P. Cartier, B. Julia, P. Moussa and P. Vanhove (eds.) Frontiers in Number Theory, Physics and Geometry, Vol 1. Springer, Berlin (2006), pp. 403–437.
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Hubert, P., Marchese, L. & Ulcigrai, C. Lagrange Spectra in Teichmüller Dynamics via Renormalization. Geom. Funct. Anal. 25, 180–255 (2015). https://doi.org/10.1007/s00039-015-0321-z
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DOI: https://doi.org/10.1007/s00039-015-0321-z