The purpose of this paper is to present some results and observations regarding the
beta transformations introduced by Rényi [8]. Given , the
beta transformation is defined for by . Parry [7] defined to be a beta number if the orbit
is finite. If for some n, then is a simple
beta number. If is a beta number which is not simple, then there is some
smallest (the preperiod length) and (the period
length) for which . For a simple beta number we define m = 0
and p to be the smallest integer with . Notice that m+p is the size
of the orbit .