Random N-Continued Fraction Expansions

Abstract

The N-continued fraction expansion is a generalization of the regular continued fraction expansion, where the digits 1 in the numerators are replaced by the natural number N. Each real number has uncountably many expansions of this form. In this article we focus on the case N=2, and we consider a random algorithm that generates all such expansions. This is done by viewing the random system as a dynamical system, and then using tools from ergodic theory to analyse these expansions. In particular, we use a recent Theorem of Inoue (2012) to prove the existence of an invariant measure of product type whose marginal in the second coordinate is absolutely continuous with respect to Lebesgue measure. Also some dynamical properties of the system are shown and the asymptotic behaviour of such expansions is investigated. Furthermore, we show that the theory can be extended to the random 3-continued fraction expansion.