A dynamical approach to \(N\)-continued fraction expansions

Karma Dajani (University of Utrecht)

Frank Adams Room 1, Alan Turing Building,

Recently (2008), Edward Burger and his co-authors introduced a new class of continued fraction algorithms, the so called \(N\)-continued fraction of the form

\[ x=n_0+\frac{N}{n_1+\frac{N}{n_2+\ddots+ \frac{N}{n_k+\ddots}}} = [n_0;n_1,n_2,\dots ,n_k,\cdots ]_N \]

where \(N, d_i\in\mathbb{Z}\), \(N, d_i\neq 0\).  They showed that for every quadratic irrational number $x$ there exist infinitely many eventually periodic \(N\)-continued fractions with period-length 1. In 2011, Maxwell Anselm and Steven Weintraub studied further the properties of \(N\)-continued fractions.  One nice result they obtained is that for \(N\ge 2\), every \(x\) between 0 and \(N\) has uncountably many \(N\)-continued fractions. In this talk we will give a dynamical approach  to \(N\)-continued fraction expansions. Due to this approach, the remarkable above mentioned result by Anselm and Weintraub is immediately obvious. We also give the ergodic properties of various subclasses of \(N\)-continued fraction expansions.

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