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 exactly, we obtain a Gauss-Kuzmin theorem related to the natural extension of the measure-dynamical system corresponding 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.