Random \(N\)-continued fraction expansions
\textit{E. B. Burger} et al. [J. Number Theory 128, No. 1, 144--153 (2008; Zbl 1210.11078)] introduced the \(N\)-continued expansion, showing that for any \(N>1\) any \(x\in\mathbb{R}\) has a representation of the form \(x=d_0+\frac{N}{d_1+\frac{N}{d_2+\cdots}}\), in which the usual continued fraction has the repeated numerator~\(1\) replaced by \(N\). ~~\textit{K. Dajani} et al. [J. Number Theory 133, No. 9, 3183--3204 (2013; Zbl 1325.11066)] showed how this expansion may be associated to a dynamical system, and used this dynamical approach to recover an earlier result of \textit{M. Anselm} and \textit{S. H. Weintraub} [J. Number Theory 131, No. 12, 2442--2460 (2011; Zbl 1263.11007)] that every \(x\in\mathbb{R}\) has infinitely many such expansions and found invariant measures for several dynamical systems generating \(N\)-continued fraction expansions. Here \(N\)-continued fraction expansions generated by a random dynamical system that randomizes across all such expansions is studied for \(N=2\) and \(3\). The approach is to use ergodic theory to study the random system as a measure-theoretic dynamical system, in particular using work of \textit{T. Inoue} [Stud. Math. 208, No. 1, 11--29 (2012; Zbl 1267.37054)] to prove the existence a measure of product type. Dynamical properties of the system are established, allowing asymptotic behaviour of the expansions to be studied.
scientific article; zbMATH DE number 6840976