Random Continued Fraction Expansions

Abstract

A 2-continued fraction expansion is a generalisation of the regular continued fraction expansion, where the digits 1 in the numerators are replaced by the natural number 2. Each real number has uncountably many expansions of this form. In this thesis 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 in 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 on the unit interval. 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 3-random continued fraction expansion.