## Random Continued Fraction Expansions

##### Summary

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.