Nearest lambda_q-multiple fractions

TL;DR

This paper studies the properties of nearest lambda_q--multiple continued fractions related to Hecke triangle groups, generalizing previous results and connecting them to geodesic flows on associated surfaces.

Contribution

It generalizes Hurwitz's result to arbitrary q and links continued fractions to geodesic flows on Hecke surfaces.

Findings

Established the conjugacy of interval maps for different q values.

Connected continued fractions to geodesic flow Poincare maps.

Constructed transfer operators for these flows.

Abstract

We discuss the nearest lambda_q--multiple continued fractions and their duals for lambda_q = 2 cos(pi/q) which are closely related to the Hecke triangle groups G_q, q=3,4,... . They have been introduced in the case q=3 by Hurwitz and for even q by Nakada. These continued fractions are generated by interval maps f_q respectively f_q^* which are conjugate to subshifts over infinite alphabets. We generalize to arbitrary q a result of Hurwitz concerning the G_q-- and f_q-equivalence of points on the real line. The natural extension of the maps f_q and f_q^* can be used as a Poincare map for the geodesic flow on the Hecke surfaces G_q\H and allows to construct the transfer operator for this flow.