Beyond Expansion III: Reciprocal Geodesics

TL;DR

This paper proves the existence of infinitely many reciprocal, low-lying, and fundamental closed geodesics on the modular surface using advanced dispersion methods in thin semigroups.

Contribution

It introduces a novel application of dispersion techniques to establish the abundance of reciprocal geodesics on the modular surface.

Findings

Infinitely many reciprocal closed geodesics exist on the modular surface.

These geodesics are low-lying and fundamental.

The method combines ideas from previous parts using dispersion in bilinear forms.

Abstract

We prove the existence of infinitely many low-lying and fundamental closed geodesics on the modular surface which are reciprocal, that is, invariant under time reversal. The method combines ideas from Parts I and II of this series, namely the dispersion method in bilinear forms, as applied to thin semigroups coming from restricted continued fractions.