Distribution in the unit tangent bundle of the geodesics of given type

TL;DR

This paper proves that geodesics of a fixed type in negatively curved surfaces become evenly distributed in the unit tangent bundle according to a specific measure, revealing properties that distinguish different hyperbolic surfaces.

Contribution

It establishes asymptotic equidistribution of geodesics of fixed type in the unit tangent bundle and analyzes the properties of the associated measure.

Findings

Geodesics of fixed type are asymptotically equidistributed in the unit tangent bundle.

The measure $rak{m}^S$ distinguishes between different hyperbolic surfaces.

Properties of the measure relate to the surface's geometry.

Abstract

Recall that two geodesics in a negatively curved surface $S$ are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure $\mathfrak{m}^S$ on $T^1S$. We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.