## Typical geodesics on flat surfaces

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- by Klaus Dankwart PDF
- Conform. Geom. Dyn.
**15**(2011), 188-209 Request permission

## 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

- Martin R. Bridson and André Haefliger,
*Metric spaces of non-positive curvature*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR**1744486**, DOI 10.1007/978-3-662-12494-9 - Marc Bourdon,
*Structure conforme au bord et flot géodésique d’un $\textrm {CAT}(-1)$-espace*, Enseign. Math. (2)**41**(1995), no. 1-2, 63–102 (French, with English and French summaries). MR**1341941** - Riccardo Benedetti and Carlo Petronio,
*Lectures on hyperbolic geometry*, Universitext, Springer-Verlag, Berlin, 1992. MR**1219310**, DOI 10.1007/978-3-642-58158-8 - Yitwah Cheung,
*Hausdorff dimension of the set of nonergodic directions*, Ann. of Math. (2)**158**(2003), no. 2, 661–678. With an appendix by M. Boshernitzan. MR**2018932**, DOI 10.4007/annals.2003.158.661 - Yitwah Cheung, Pascal Hubert, and Howard Masur,
*Dichotomy for the Hausdorff dimension of the set of nonergodic directions*, Invent. Math.**183**(2011), no. 2, 337–383. MR**2772084**, DOI 10.1007/s00222-010-0279-2 - Michel Coornaert,
*Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov*, Pacific J. Math.**159**(1993), no. 2, 241–270 (French, with French summary). MR**1214072**, DOI 10.2140/pjm.1993.159.241 - Michel Coornaert and Athanase Papadopoulos,
*Symbolic dynamics and hyperbolic groups*, Lecture Notes in Mathematics, vol. 1539, Springer-Verlag, Berlin, 1993. MR**1222644** - 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**(1994), no. 2, 883–898 (French). MR**1207579**, DOI 10.1090/S0002-9947-1994-1207579-2 - Michel Coornaert and Athanase Papadopoulos,
*Upper and lower bounds for the mass of the geodesic flow on graphs*, Math. Proc. Cambridge Philos. Soc.**121**(1997), no. 3, 479–493. MR**1434656**, DOI 10.1017/S0305004196001478 - Klaus Dankwart. Volume entropy and the Gromov boundary of flat surfaces.
*ArXiv e-prints*, January 2011. - Moon Duchin, Christopher J. Leininger, and Kasra Rafi,
*Length spectra and degeneration of flat metrics*, Invent. Math.**182**(2010), no. 2, 231–277. MR**2729268**, DOI 10.1007/s00222-010-0262-y - M. Gromov,
*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, DOI 10.1007/978-1-4613-9586-7_{3} - Eberhard Hopf,
*Ergodic theory and the geodesic flow on surfaces of constant negative curvature*, Bull. Amer. Math. Soc.**77**(1971), 863–877. MR**284564**, DOI 10.1090/S0002-9904-1971-12799-4 - John Hamal Hubbard,
*Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1*, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR**2245223** - Vadim A. Kaimanovich,
*Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces*, J. Reine Angew. Math.**455**(1994), 57–103. MR**1293874**, DOI 10.1515/crll.1994.455.57 - Linda Keen,
*Collars on Riemann surfaces*, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 263–268. MR**0379833** - Howard Masur,
*Interval exchange transformations and measured foliations*, Ann. of Math. (2)**115**(1982), no. 1, 169–200. MR**644018**, DOI 10.2307/1971341 - Howard Masur,
*Closed trajectories for quadratic differentials with an application to billiards*, Duke Math. J.**53**(1986), no. 2, 307–314. MR**850537**, DOI 10.1215/S0012-7094-86-05319-6 - Howard Masur,
*The growth rate of trajectories of a quadratic differential*, Ergodic Theory Dynam. Systems**10**(1990), no. 1, 151–176. MR**1053805**, DOI 10.1017/S0143385700005459 - Howard Masur,
*Hausdorff dimension of the set of nonergodic foliations of a quadratic differential*, Duke Math. J.**66**(1992), no. 3, 387–442. MR**1167101**, DOI 10.1215/S0012-7094-92-06613-0 - Howard Masur,
*Ergodic theory of translation surfaces*, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 527–547. MR**2186247**, DOI 10.1016/S1874-575X(06)80032-9 - Yair N. Minsky,
*Harmonic maps, length, and energy in Teichmüller space*, J. Differential Geom.**35**(1992), no. 1, 151–217. MR**1152229** - Kasra Rafi,
*Thick-thin decomposition for quadratic differentials*, Math. Res. Lett.**14**(2007), no. 2, 333–341. MR**2318629**, DOI 10.4310/MRL.2007.v14.n2.a14 - Kurt Strebel,
*Quadratic differentials*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR**743423**, DOI 10.1007/978-3-662-02414-0 - Dennis Sullivan,
*The density at infinity of a discrete group of hyperbolic motions*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 171–202. MR**556586**, DOI 10.1007/BF02684773 - W.P. Thurston.
*The Geometry and Topology of Three-manifolds*. Princeton University, 1980. - Ya. B. Vorobets,
*Plane structures and billiards in rational polygons: the Veech alternative*, Uspekhi Mat. Nauk**51**(1996), no. 5(311), 3–42 (Russian); English transl., Russian Math. Surveys**51**(1996), no. 5, 779–817. MR**1436653**, DOI 10.1070/RM1996v051n05ABEH002993

## Additional Information

**Klaus Dankwart**- Affiliation: Vorgebirgsstrasse 80, 53119 Bonn, Germany
- Email: kdankwart@googlemail.com
- 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
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2869013