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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
Published electronically: November 17, 2011
MathSciNet review: 2869013
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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 $\exp (-e(S)l(c))$.

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Additional Information

Klaus Dankwart
Affiliation: Vorgebirgsstrasse 80, 53119 Bonn, Germany

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