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

Abstract

We consider a family {T N : N ≥ 1} of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from T N , we solve its Gauss-Kuzmin-type problem by applying the theory of random systems with complete connections by Iosifescu. Mathematics Subject Classifications (2010). 11J70, 11K50

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

“…Our study is a continuation of the papers [11] and [12], where the second author discussed some metric properties of N -continued fraction expansions. Moreover, he investigated the associated Perron-Frobenius operator and proved a Gauss-Kuzmin theorem for N -continued fractions by applying the method of random systems with complete connections (RSCC) by Iosifescu [8].…”

“…, a 2 , a 1 + t] N , n ≥ 2, while s 1,t = N/(a 1 + t), t ∈ I. These facts lead us to the random system with complete connections [11] {(I, B I ), (Λ, P(Λ)), u, V }, where P(Λ) is the power set of Λ,…”

“…In this section, we give a Gauss-Kuzmin theorem related to the natural extension I 2 , B 2 I , G N , T N . To solve the problem, we need a version of the Gauss-Kuzmin theorem for T N different from that presented by the second author in [11]. The problem of finding the asymptotic behaviour of T −n N (A) as n → ∞, A ∈ B I , represents the Gauss-Kuzmin problem for N -continued fraction expansions.…”

Invariant measures for continued fraction algorithms with finitely many digits

“…This method is based on a Gauss-Kuzmin-Lévy theorem. For greedy N -expansions this theorem is proved by Dan Lascu in [13]. The method yields smoother results than by simulating in the classical way (looking at the histogram of the orbit of a typical point as described in Choe's book [14], and used in his papers [12,15]).…”

A generalization of Gauss-Kuzmin-Lévy theorem

“…where {x} is the fractional part of x. Such transformations were first introduced in [3] (the associated continued fractions had appeared in [2]) and also studied in [7][8] [11]. For every p, T p has a unique absolutely continuous ergodic invariant measure dµ p (x) = 1 ln(p + 1) − ln p…”