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Now, let be the probability that . Then the probability that the points are distinct is

This suggests that almost all orbits are finite if **n = 4,6** while a positive
proportion are infinite if .

We can distinguish the cases **n = 4** and **n = 6** by considering the expected
size (**D = m+p**) of the orbit. Using the familiar fact that , we compute

Remark 6.2 For **d = 4**, we have shown rigorously in [3] that
. The largest value of for
with a fixed value of is attained by the with minimal
polynomial . By (2.4), **P** has discriminant
for large , so that from (6.4),
. Thus (6.7) predicts a smaller value for than
actually attained but it does predict that large values of occur for small
values of and vice versa.

Remark 6.3 For **d = 6**, the data tends to confirm a direct relationship
between the sizes of and , although not quite as dramatic as (6.8)
would suggest. There are some exceptions as well: for example if , then is quite small, so is
large, but is not exceptionally large. On the other hand, the
example with only has , which
is not very large (although it * is* the largest value of by far in
Table 3, i.e., for ). This is perhaps to be expected, given the number
of unsupported assumptions we have had to make to arrive at (6.8).

Perhaps the most questionable argument is the deduction of the exponent
in section 6.4. Let denote the maximum modulus of the conjugates of ,
i.e., the largest modulus of the eigenvalues of the matrix discussed in section 5.2.
A fit of a power curve to data from the first **20000** values of for the case
suggests an exponent for this example. A
plot of versus **n** in this range suggests some sort of random process but
perhaps not one well described by the model of section 6.4. Note that, for **d = 6**, we only
need to apply the birthday paradox argument of section 6.3, so if the growth
observed for were to persist, then section 6.3 would predict a periodic beta
expansion for this .

To test this, let us consider the behavior of the record values of for large
**n** still for this . So, let denote the largest value of for . For , , we find that

Remark 6.4 Our computations have concentrated on the question of whether
Salem numbers are beta numbers and not on Schmidt's more general conjecture that, for
every Salem number and every rational , the orbit
should be finite. Our heuristic arguments apply equally well to this conjecture. If , then the iterates lie in , and hence we need only
replace the lattice by in the arguments of section 6.1 to section 6.6. The
only change is that the density of the points of in is
times as large as the density of the points of . Thus, the qualitative
predictions of section 6.1 to section 6.4 are unchanged, i.e., we predict that Schmidt's conjecture
is true for Salem numbers of degree **4** and **6** but not for higher degrees. The main
change in the more quantitative analysis of section 6.5 and section 6.6 is that must be
replaced by . Thus, section 6.6 would suggest that, for fixed , the size
of the orbit should increase with **q** as predicted by (6.7) and (6.8).
That is, for **d = 4** the size of the orbit should be roughly proportional to , and
for **d = 6** we should expect unusually large orbits as soon as .
It would be interesting to test these predictions by further computation.

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