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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 ).

### [Proof]

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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