Ergodicity of N-continued fraction expansions

Recently, Edward Burger and his co-authors introduced and studied in Burger et al. (2008) [3] a new class of continued fraction algorithms. In particular they showed that for every quadratic irrational number x there exist infinitely many eventually periodic N-expansions with period-length 1; see also Komatsu (2009) [10] for related properties. In 2011, Maxwell Anselm and Steven Weintraub further studied the properties of N-expansions in Anselm and Weintraub (2011) [2]. One nice result they obtained is that every x between 0 and N has uncountably many N  -expansions for each integer N⩾2N⩾2. In this paper we will reprove this result and from this we study the ergodic properties of various subclasses of N-expansions.