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 .
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