Metric and Arithmetic Properties of Mediant-rosen Maps

We define maps which induce mediant convergents of Rosen continued fractions and discuss arithmetic and metric properties of mediant convergents. In particular, we show equality of the ergodic theoretic Lenstra constant with the arithmetic Legendre constant for each of these maps. This value is sufficiently small that the mediant Rosen convergents directly determine the Hurwitz constant of Diophantine approximation of the underlying Fuchsian group. We thus succeed in giving a continued fractions based verification of these Hurwitz values.