Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Typical geodesics on flat surfaces


Author: Klaus Dankwart
Journal: Conform. Geom. Dyn. 15 (2011), 188-209
MSC (2010): Primary 30F30, 37E35; Secondary 30F60
DOI: https://doi.org/10.1090/S1088-4173-2011-00234-7
Published electronically: November 17, 2011
MathSciNet review: 2869013
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate typical behavior of geodesics on a closed flat surface $ S$ of genus $ g\geq 2$. We compare the length quotient of long arcs in the same homotopy class with fixed endpoints for the flat and the hyperbolic metric in the same conformal class. This quotient is asymptotically constant $ F$ a.e. We show that $ F$ is bounded from below by the inverse of the volume entropy $ e(S)$. Moreover, we construct a geodesic flow together with a measure on $ S$ which is induced by the Hausdorff measure of the Gromov boundary of the universal cover. Denote by $ e(S)$ the volume entropy of $ S$ and let $ c$ be a compact geodesic arc which connects singularities. We show that a typical geodesic passes through $ c$ with frequency that is comparable to $ \exp (-e(S)l(c))$. Thus a typical bi-infinite geodesic contains infinitely many singularities, and each geodesic between singularities $ c$ appears infinitely often with a frequency proportional to $ \exp (-e(S)l(c))$.


References [Enhancements On Off] (What's this?)

  • [BH99] Martin R. Bridson and André Haefliger.
    Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],
    Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • [Bou95] Marc Bourdon.
    Structure conforme au bord et flot géodésique d'un $ {\rm CAT}(-1)$-espace.
    Enseign. Math. (2) 41(1-2):63-102, 1995. MR 1341941 (96f:58120)
  • [BP92] Riccardo Benedetti and Carlo Petronio.
    Lectures on hyperbolic geometry,
    pages xiv+330, 1992. MR 1219310 (94e:57015)
  • [Che03] Yitwah Cheung.
    Hausdorff dimension of the set of nonergodic directions.
    Ann. of Math. (2) 158(2):661-678, 2003.
    With an appendix by M. Boshernitzan. MR 2018932 (2004k:37069)
  • [CHM10] Yitwah Cheung, Pascal Hubert, and Howard Masur.
    Dichotomy for the Hausdorff dimension of the set of nonergodic directions.
    Inventiones Mathematicae, pages 1-47, 2010.
    10.1007/s00222-010-0279-2. MR 2772084
  • [Coo93] Michel Coornaert.
    Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov,
    Pacific J. Math. 159(2):241-270, 1993. MR 1214072 (94m:57075)
  • [CP93] Michel Coornaert and Athanase Papadopoulos.
    Symbolic dynamics and hyperbolic groups, volume 1539 of Lecture Notes in Mathematics,
    Springer-Verlag, Berlin, 1993. MR 1222644 (94d:58054)
  • [CP94] M. Coornaert and A. Papadopoulos.
    Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres,
    Trans. Amer. Math. Soc. 343(2):883-898, 1994. MR 1207579 (94h:58131)
  • [CP97] Michel Coornaert and Athanase Papadopoulos.
    Upper and lower bounds for the mass of the geodesic flow on graphs,
    Math. Proc. Cambridge Philos. Soc. 121(3):479-493, 1997. MR 1434656 (98d:58140)
  • [Dan11] Klaus Dankwart.
    Volume entropy and the Gromov boundary of flat surfaces.
    ArXiv e-prints, January 2011.
  • [DLR10] Moon Duchin, Christopher Leininger, and Kasra Rafi.
    Length spectra and degeneration of flat metrics,
    Inventiones Mathematicae 182:231-277, 2010.
    10.1007/s00222-010-0262-y. MR 2729268
  • [Gro87] M. Gromov.
    Hyperbolic groups.
    In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75-263. Springer, New York, 1987. MR 919829 (89e:20070)
  • [Hop71] Eberhard Hopf.
    Ergodic theory and the geodesic flow on surfaces of constant negative curvature,
    Bull. Amer. Math. Soc. 77:863-877, 1971. MR 0284564 (44:1789)
  • [Hub06] John Hamal Hubbard.
    Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1,
    pages xx+459, 2006,
    Teichmüller theory, with contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra and with forewords by William Thurston and Clifford Earle. MR 2245223 (2008k:30055)
  • [Kai94] Vadim A. Kaimanovich.
    Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces,
    J. Reine Angew. Math. 455:57-103, 1994. MR 1293874 (95g:58130)
  • [Kee74] Linda Keen.
    Collars on Riemann surfaces,
    pages 263-268. Ann. of Math. Studies, No. 79, 1974. MR 0379833 (52:738)
  • [Mas82] Howard Masur.
    Interval exchange transformations and measured foliations,
    Ann. of Math. (2) 115(1):169-200, 1982. MR 644018 (83e:28012)
  • [Mas86] H. Masur.
    Closed trajectories for quadratic differentials with an application to billiards,
    Duke Math. J. 53(2):307-314, 1986. MR 850537 (87j:30107)
  • [Mas90] Howard Masur.
    The growth rate of trajectories of a quadratic differential,
    Ergodic Theory Dynam. Systems 10(1):151-176, 1990. MR 1053805 (91d:30042)
  • [Mas92] Howard Masur.
    Hausdorff dimension of the set of nonergodic foliations of a quadratic differential,
    Duke Math. J. 66(3):387-442, 1992. MR 1167101 (93f:30045)
  • [Mas06] Howard Masur.
    Ergodic theory of translation surfaces.
    In Handbook of dynamical systems. Vol. 1B, pages 527-547, Elsevier B. V., Amsterdam, 2006. MR 2186247 (2006i:37012)
  • [Min92] Yair N. Minsky.
    Harmonic maps, length, and energy in Teichmüller space,
    J. Differential Geom. 35(1):151-217, 1992. MR 1152229 (93e:58041)
  • [Raf07] Kasra Rafi.
    Thick-thin decomposition for quadratic differentials,
    Math. Res. Lett. 14(2):333-341, 2007. MR 2318629 (2008g:30035)
  • [Str84] Kurt Strebel.
    Quadratic differentials, volume 5 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],
    Springer-Verlag, Berlin, 1984. MR 743423 (86a:30072)
  • [Sul79] Dennis Sullivan.
    The density at infinity of a discrete group of hyperbolic motions.
    Inst. Hautes Études Sci. Publ. Math. (50):171-202, 1979. MR 556586 (81b:58031)
  • [Thu80] W.P. Thurston.
    The Geometry and Topology of Three-manifolds.
    Princeton University, 1980.
  • [Vor96] Ya. B. Vorobets.
    Plane structures and billiards in rational polygons: the Veech alternative,
    Uspekhi Mat. Nauk 51(5(311)):3-42, 1996. MR 1436653 (97j:58092)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30F30, 37E35, 30F60

Retrieve articles in all journals with MSC (2010): 30F30, 37E35, 30F60


Additional Information

Klaus Dankwart
Affiliation: Vorgebirgsstrasse 80, 53119 Bonn, Germany
Email: kdankwart@googlemail.com

DOI: https://doi.org/10.1090/S1088-4173-2011-00234-7
Received by editor(s): February 20, 2011
Published electronically: November 17, 2011
Additional Notes: This research was supported by Bonn International Graduate School in Mathematics
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society