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If is a Salem number of degree **d = 2s**, then it has two real conjugates,
and , and **s-1** pairs of complex conjugates
for . All the numbers lie in the set . As is familiar from the geometry of numbers
[2], p.96, we can think of as a lattice in
defined by mapping onto the point ,
where denotes the conjugate of corresponding to the conjugation
.
The determinant of [2], p.99 is
, since there are **d-2** nonreal
conjugates, where denotes not the discriminant of the field , but rather the discriminant of the order i.e.,
the discriminant of the minimal polynomial **P** of . Thus, the number of points
of in a large cube of volume **V** is asymptotically .

Now, consider the iterates as points in the lattice
. By definition, . The conjugate of
corresponding to is , which satisfies
. For a typical conjugate with , the corresponding conjugate of is . However, it is reasonable to expect that may be
for some . In fact, it is plausible that , as we
argue in section 6.4 below. The following result contains the only completely rigorous argument
of this section.

### Proposition 6.1

Suppose that , and that for all conjugates , for some . Then is
a beta number (and hence ).

Remark 6.1 It is not clear how one would prove that
with for some given . The main virtue of Proposition 6.1 is that
it shows that there is a threshold for the rate of growth of which must be
exceeded before nonperiodicity is possible.

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