One can easily show that for any given length k there are only a finite number of integers n that are periodic under iteration by T with period k, in fact at most such integers, as observed by Böhm and Sontacchi [13]. To see this, substitute the equation (2.4) into


to obtain the equation


There are only choices for the 0-1 vector , and for each choice of the equation (2.24) determines a unique rational solution . Consequently there are at most solutions to (2.23). Böhm and Sontacchi also noted that this gives an (inefficient) finite procedure for deciding if there are any cycles of a given length k, as follows: Determine the rational number for each of the vectors , and for each which is an integer test if (2.23) holds.
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