Here is an outline of the probabalistic argument which is given in more detail in § 6 of [4]. For simplicity, we present the argument only for but it clearly applies to general rational . Writing for the size of the orbit , the argument correctly predicts that will be small relative to for Salem numbers of degree 4. It predicts that will be finite for almost all Salem numbers of degree 6 but that can be arbitrarily large even for small . Moreover, it predicts that for each fixed even degree there should be a positive proportion of Salem numbers of degree d for which , as well as a positive proportion for which is finite. This latter prediction has not yet been tested by extensive experiment.
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