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

Abstract

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.

A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

“…Apart from the RCF expansions, there is a wide variety of continued fraction expansions. Here, we mention only a few of the expansions studied from the metrical point of view by the authors over time, namely Chan's continued fractions [14,15], θ−expansions [16,17], N-continued fraction expansions [18,19], and Rényi-type continued fraction expansions [20][21][22].…”

“…We now check Rényi condition (8). We use directly, without mentioning them here, some properties proved in [19]. Thus, we have…”

“…There are still many open questions closely related to this problem. On the other hand, to extend our references list [14][15][16][17][18][19][20][21][22], we consider the opportunity of starting new investigations such as the efficiency of pairs of maps for which the generating partition has finite entropy. In these circumstances, we review sufficient conditions on the latter to belong to a class of number-theoretic fibered maps for which the generating partition has finite entropy.…”