# Measuring geodesics’ aperiodicity

Mätzener, Anna. Measuring geodesics’ aperiodicity. 2011, University of Zurich, Faculty of Science.

## Abstract

Aperiodicity can arise both in the setting of sequences over a finite alphabet, and that of geodesics on a compact Riemannian surface. In both cases, aperiodicity itself provides no means to measure and compare different aperiodic objects one to another. For sequences the notion of -aperiodicity, by the function , provides a means for this. The aim of this thesis was to find an analogon in the setting of geodesics. This was done by defining f-aperiodicity of geodesics. The existence of f-aperiodic geodesics was proven for a very specific setting, namely that of a quotient of the hyperbolic surface of H. This quotient was chosen in a specific way, such that a -aperiodic sequence could be chosen as the origin in the construction of the geodesic. Furthermore, this led to an easy way to define a flow-invariant subset of the unit tangent bundle of the compact Riemannian surface.

## Abstract

Aperiodicity can arise both in the setting of sequences over a finite alphabet, and that of geodesics on a compact Riemannian surface. In both cases, aperiodicity itself provides no means to measure and compare different aperiodic objects one to another. For sequences the notion of -aperiodicity, by the function , provides a means for this. The aim of this thesis was to find an analogon in the setting of geodesics. This was done by defining f-aperiodicity of geodesics. The existence of f-aperiodic geodesics was proven for a very specific setting, namely that of a quotient of the hyperbolic surface of H. This quotient was chosen in a specific way, such that a -aperiodic sequence could be chosen as the origin in the construction of the geodesic. Furthermore, this led to an easy way to define a flow-invariant subset of the unit tangent bundle of the compact Riemannian surface.