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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Typical geodesics on flat surfaces
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by Klaus Dankwart PDF
Conform. Geom. Dyn. 15 (2011), 188-209 Request permission


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 $\exp (-e(S)l(c))$.
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Additional Information
  • Klaus Dankwart
  • Affiliation: Vorgebirgsstrasse 80, 53119 Bonn, Germany
  • Email:
  • 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:
  • MathSciNet review: 2869013