Metric and arithmetic properties of mediant-Rosen maps

TL;DR

This paper investigates the properties of mediant-Rosen maps, using continued fractions and ergodic theory to analyze Diophantine approximation and verify Hurwitz values for Hecke triangle groups.

Contribution

It provides a continued fractions approach to verify Hurwitz values and demonstrates ergodic theory's role in Diophantine approximation for Rosen continued fractions.

Findings

Verification of Hurwitz values for Hecke triangle groups

Diophantine approximation by mediant convergents suffices for key values

Completes Lehner's program using ergodic theory

Abstract

A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.