Dependence with complete connections and the Gauss–Kuzmin theorem for N-continued fractions

We consider a family {TN:N≥1}{TN:N≥1} of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from TNTN, we solve its Gauss–Kuzmin-type problem by applying the theory of random systems with complete connections by Iosifescu.