Flat Surfaces

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. Such flat surfaces are naturally organized into families which appear to be isomorphic to moduli spaces of holomorphic one-forms. One can obtain much information about the geometry and dynamics of an individual flat surface by studying both its orbit under the Teichmüller geodesic flow and under the linear group action. In particular, the Teichmüller geodesic flow plays the role of a time acceleration machine (renormalization procedure) which allows to study the asymptotic behavior of interval exchange transformations and of surface foliations. This survey is an attempt to present some selected ideas, concepts and facts in Teichmüller dynamics in a playful way. Frontiers in Number Theory, Physics, and Geometry Vol.I, P. Cartier; B. Julia; P. Moussa; P. Vanhove (Editors), Springer Verlag, 2006, 439–586.

[1]  The theta characteristic of a branched covering , 2003, math/0312186.

[2]  G. Forni.  Chapter 8 - On the Lyapunov Exponents of the Kontsevich–Zorich Cocycle , 2006 .

[3]  Pascal Hubert,et al.  Problems on billiards, flat surfaces and translation surfaces , 2006 .

[4]  Steven P. Kerckhoff,et al.  Ergodicity of billiard flows and quadratic differentials , 1986 .

[5]  L. Barreira,et al.  Lyapunov Exponents and Smooth Ergodic Theory , 2002 .

[6]  H. Masur,et al.  Billiards in rectangles with barriers , 2001, math/0107204.

[7]  On triangular billiards , 2001 .

[8]  Thomas A. Schmidt,et al.  Veech groups and polygonal coverings , 2000 .

[9]  J. Milnor.  Remarks Concerning Spin Manifolds , 1965 .

[10]  Martin Moeller.  Teichmueller curves, Galois actions and GT-relations , 2003, math/0311308.

[11]  Stefano Marmi,et al.  Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps , 2004, 1407.1776.

[12]  Michael Atiyah,et al.  Riemann surfaces and spin structures , 1971 .

[13]  W. Veech.  Gauss measures for transformations on the space of interval exchange maps , 1982 .

[14]  W. Veech.  Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , 1989 .

[15]  C. McMullen.  Teichmüller curves in genus two: Torsion divisors and ratios of sines , 2006 .

[16]  Anton Zorich,et al.  Connected components of the moduli spaces of Abelian differentials with prescribed singularities , 2002 .

[17]  Xoyo-L.  CLOSURES OF TOTALLY GEODESIC IMMERSIONS IN MANIFOLDS OF CONSTANT NEGATIVE CURVATURE , 2007 .

[18]  C. McMullen.  Teichmüller geodesics of infinite complexity , 2003 .

[19]  Noncongruence subgroups in H(2) , 2004, math/0410595.

[20]  Anton Zorich,et al.  Finite Gauss measure on the space of interval exchange transformations , 1996 .

[21]  E. Gutkin,et al.  Affine mappings of translation surfaces: geometry and arithmetic , 2000 .

[22]  Thomas A. Schmidt,et al.  INFINITELY GENERATED VEECH GROUPS , 2004 .

[23]  C. McMullen.  Teichmüller curves in genus two: Discriminant and spin , 2005 .

[24]  W. Veech.  THE METRIC THEORY OF INTERVAL EXCHANGE TRANSFORMATIONS II. APPROXIMATION BY PRIMITIVE INTERVAL EXCHANGES , 1984 .

[25]  Closures of totally geodesic immersions into locally symmetric spaces of noncompact type , 1999 .

[26]  H. Herzel.  Chaotic Evolution and Strange Attractors , 1991 .

[27]  S.P.Novikov,et al.  Topological Phenomena in Normal Metals , 1997, cond-mat/9709007.

[28]  John D. Fay.  Theta Functions on Riemann Surfaces , 1973 .

[29]  H. Masur.  The extension of the Weil-Petersson metric to the boundary of Teichmuller space , 1976 .

[30]  V. Arnold.  SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS , 1963 .

[31]  Martin Möller.  Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve , 2004, math/0410262.

[32]  C. McMullen.  Dynamics of SL 2 ( R ) over moduli space in genus two , 2003 .

[33]  Anton Zorich,et al.  Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants , 2002 .

[34]  Marcelo Viana,et al.  Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture , 2005, math/0508508.

[35]  D. Mumford.  Theta characteristics of an algebraic curve , 1971 .

[36]  S. Kerckhoff.  The asymptotic geometry of teichmuller space , 1980 .

[37]  Richard Kenyon,et al.  Billiards on rational-angled triangles , 2000 .

[38]  A. Haefliger,et al.  Group theory from a geometrical viewpoint , 1991 .

[39]  A. Zorich.  Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials , 2002 .

[40]  M. Keane,et al.  Non-ergodic interval exchange transformations , 1977 .

[41]  Dennis Johnson.  Spin Structures and Quadratic forms on Surfaces , 1980 .

[42]  Jean-Christophe Yoccoz,et al.  Continued Fraction Algorithms for Interval Exchange Maps: an Introduction ? , 2003 .

[43]  Thomas A. Schmidt,et al.  Invariants of translation surfaces , 2001 .

[44]  Martin Moeller.  Variations of Hodge structures of a Teichmüller curve , 2004, math/0401290.

[45]  Serge Tabachnikov,et al.  Rational billiards and flat structures , 2002 .

[46]  Mark Pollicott,et al.  Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds , 1993 .

[47]  H. Masur,et al.  Asymptotic formulas on flat surfaces , 2001, Ergodic Theory and Dynamical Systems.

[48]  H. Masur.  Closed trajectories for quadratic differentials with an application to billiards , 1986 .

[49]  Geometry of infinitely generated Veech groups , 2004, math/0410132.

[50]  Giovanni Forni.  Deviation of ergodic averages for area-preserving flows on surfaces of higher genus , 2002 .

[51]  A. N. Zemlyakov,et al.  Topological transitivity of billiards in polygons , 1975 .

[52]  P. Levy,et al.  Sur le développement en fraction continue d'un nombre choisi au hasard , 1936 .

[53]  Multiple Saddle Connections on Flat Surfaces and Principal Boundary of the Moduli Spaces of Quadrati , 2004 .

[54]  Gérard Rauzy,et al.  Échanges d'intervalles et transformations induites , 1979 .

[55]  S. Kerckhoff.  Simplicial systems for interval exchange maps and measured foliations , 1985, Ergodic Theory and Dynamical Systems.

[56]  W. Veech.  THE METRIC THEORY OF INTERVAL EXCHANGE TRANSFORMATIONS I. GENERIC SPECTRAL PROPERTIES , 1984 .

[57]  H. Masur.  Interval Exchange Transformations and Measured Foliations , 1982 .

[58]  C. McMullen.  Billiards and Teichmüller curves on Hilbert modular surfaces , 2003 .

[59]  I. Dynnikov.  Surfaces in 3-Torus: Geometry of Plane Sections , 1998 .

[60]  Anton Zorich,et al.  Deviation for interval exchange transformations , 1997, Ergodic Theory and Dynamical Systems.

[61]  S. Novikov.  The Hamiltonian formalism and a many-valued analogue of Morse theory , 1982 .

[62]  Artur Avila,et al.  Weak mixing for interval exchange transformations and translation flows , 2004 .

[63]  C. McMullen.  Teichmüller curves in genus two: The decagon and beyond , 2005 .

[64]  Prime arithmetic Teichmuller discs in H(2) , 2004 .

[65]  D. Rudolph,et al.  Topological weak-mixing of interval exchange maps , 1997, Ergodic Theory and Dynamical Systems.

[66]  M. Boshernitzan.  A condition for minimal interval exchange maps to be uniquely ergodic , 1985 .

[67]  Yaroslav Vorobets.  Planar structures and billiards in rational polygons: the Veech alternative , 1996 .

[68]  A. Eskin,et al.  Unipotent flows on the space of branched covers of Veech surfaces , 2004, Ergodic Theory and Dynamical Systems.

[69]  Gabriela Schmithüsen,et al.  An Algorithm for Finding the Veech Group of an Origami , 2004, Exp. Math..

[70]  AN ALGORITHM FOR FINDING THE VEECH GROUP OF AN ORIGAMI , 2004 .

[71]  C. McMullen.  Prym Varieties and Teichmüller Curves , 2006 .

[72]  H. Masur.  The growth rate of trajectories of a quadratic differential , 1990, Ergodic Theory and Dynamical Systems.

[73]  H. Masur.  Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential , 1988 .

[74]  L. Bers.  Finite dimensional Teichmüller spaces and generalizations , 1981 .

[75]  A. Katok.  Interval exchange transformations and some special flows are not mixing , 1980 .