A two-dimensional Gauss—Kuzmin theorem for $N$-continued fraction expansions

A two-dimensional Gauss—Kuzmin theorem for $N$-continued fraction expansions is shown. More precisely, we obtain a Gauss—Kuzmin theorem related to the natural extension of the measure-theoretical dynamical system associated to these expansions. Then, using characteristic properties of the transition operator associated with the random system with complete connections underlying $N$-continued fractions on the Banach space of complex-valued functions of bounded variation, we derive explicit lower and upper bounds for the convergence rate of the distribution function to its limit.