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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