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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

For d =4, we thus have , so

On the other hand, if d = 6, then

i.e., , so that

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

Fitting a power curve to these 9 values (by a linear least squares fit to their logarithms) gives a curve , suggesting . On the other hand, using only gives , while the data for the 5 values is very well fitted by a curve with . Apparently, this data does not exclude or confirm a growth rate of . Another instructive example is , where . Here, is quite well fitted for by a curve with . Of course, this cannot hold for since is ultimately periodic.

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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