## 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.