The determinant of , 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.
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.