Typical geodesics on flat surfaces

TL;DR

This paper studies the behavior of typical geodesics on flat surfaces of genus at least 2, revealing how they relate to volume entropy and how often they pass through singularities, with implications for understanding their geometric and dynamical properties.

Contribution

It introduces a new measure on geodesic flow related to the Gromov boundary and establishes bounds on geodesic behavior in terms of volume entropy.

Findings

The length quotient of long arcs is asymptotically constant almost everywhere.

A typical geodesic passes through a given arc with frequency proportional to \\exp(-e(S)l(c)).

Typical bi-infinite geodesics contain infinitely many singularities.

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 $\exp(-e(S)l(c))$.