Backward continued fractions, Hecke groups and invariant measures for transformations of the interval

Abstract

AbstractWe develop a new type of backward continued fractions that can be associated to each Hecke-type group. We study its symbolic dynamics, and the corresponding interval maps and their invariant measures. These measures are infinite if and only if the corresponding groups are discrete. For the discrete Hecke groups the invariant measure is computed explicitly by studying the geodesic flow on the associated Riemann surface.