Computation of invariant densities for continued fraction algorithms

We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm that computes exact invariant densities of certain continued fraction algorithms, including classical ones and some of their extensions. For finite extensions of the classical additive algorithm with two coordinates, we provide a more precise algorithm that decides whether the invariant density is composed of rational fractions and computes it. For any finite set of quadratic numbers, we construct a continued fraction algorithm whose invariant density are rational fractions containing the quadratic numbers.