Applications of (a,b)-continued fraction transformations

We describe a general method of arithmetic coding of geodesics on the modular surface based on the study of one-dimensional Gauss-like maps associated to a two-parameter family of continued fractions introduced in [Katok and Ugarcovici. Structure of attractors for (a,b)-continued fraction transformations. J. Modern Dynamics 4 (2010), 637-691]. The finite rectangular structure of the attractors of the natural extension maps and the corresponding reduction theory play an essential role. In special cases, when an (a,b)-expansion admits a so-called dual, the coding sequences are obtained by juxtaposition of the boundary expansions of the fixed points, and the set of coding sequences is a countable sofic shift. We also prove that the natural extension maps are Bernoulli shifts and compute the density of the absolutely continuous invariant measure and the measure-theoretic entropy of the one-dimensional map.