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The probabilistic model


Here is an outline of the probabalistic argument which is given in more detail in § 6 of [4]. For simplicity, we present the argument only for but it clearly applies to general rational . Writing for the size of the orbit , the argument correctly predicts that will be small relative to for Salem numbers of degree 4. It predicts that will be finite for almost all Salem numbers of degree 6 but that can be arbitrarily large even for small . Moreover, it predicts that for each fixed even degree there should be a positive proportion of Salem numbers of degree d for which , as well as a positive proportion for which is finite. This latter prediction has not yet been tested by extensive experiment.
The heuristic argument suggests, for Salem numbers of fixed degree d, that the size of is directly related to the size of . Here denotes the discriminant of the polynomial P. We explore this connection through some of the computed examples for d = 6.

For Schmidt's more general conjecture the model predicts that if d = 4, the size of the orbit for fixed should be roughly proportional to . This has also not been extensively tested.

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 we can think of as a lattice L in defined by mapping onto the point , where denotes the conjugate of corresponding to the conjugation . The determinant of L is , since there are d-2 nonreal conjugates, where denotes the discriminant of the order i.e., the discriminant of the minimal polynomial P of . Thus, the number of points of L in a large cube of volume V is asymptotically .

Now, consider the iterates as points in the lattice L. 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, a random walk argument makes it plausible that .

For example, Figure 1 shows the first 50000 values of for the case , which suggests some sort of random process. A fit of a power curve to this data (the solid curve in Figure 1 ) suggests an exponent . However, using the data up to suggests an exponent in the range , consistent with . Figure 2 shows the case , where . For the initial transient, is well fitted for by a curve with .

The points of L lie in a slab which is the product of two intervals of bounded length, corresponding to the conjugates and , and disks of radius corresponding to the conjugates . The volume of is , which is if . contains points of L by a result of Senechal and de Bruijn [8,]. (This result was not used in [4], where the conclusion that is presented as being merely plausible. It is in fact rigorously justified). For each n, the n points lie in the set of size , so eventually there must be a coincidence between two points with distinct indices, by the box principle, and hence the orbit must be periodic. Note that, even for d = 4, this argument would require , and we expect that in fact . For d = 6, we would need , which does not seem reasonable.

If we go beyond the box principle and imagine that the points of the orbit are distributed ``randomly'' in the slab , we can allow a larger value of by making use of the ``birthday paradox''. Suppose that with , so the volume of satisfies and hence contains points of the lattice L. Now suppose that the points are randomly chosen from these points. Then the ``probability'' that these n are distinct is

as since .

For d = 4, this argument requires , which is just short of what can be proved, while for d = 6, the requirement is , which is just short of the suggested by random walk. For , however, even the most favorable assumptions would not seem to justify the expectation that the orbit should be finite.

Let us now assume that indeed , and that , where the O constant is uniform in . The region , being the product of two intervals of lengths 1 and and disks of radius , has volume . Since the points of L are distributed with density there are points of L in . If is sufficiently large, it is thus possible to push through the box principle argument for d = 4 and the birthday paradox argument for d = 6 to conclude that the orbit of should be finite. For d = 8, these assumptions lead to the conclusion that for a positive proportion of . Being more precise, one finds that , where

One can make plausible arguments that typically . Values of which are untypically small will lead to large values of .

It turns out that the cases d = 4 and d = 6 in which we expect are distinguised by the expected size of the orbit :

For d =4, the prediction is that

which is somewhat smaller than rigorously proved in [2] for . On the other hand, if d = 6, we find that

The data for described in [4] tends to confirm a direct relationship between the sizes of and , although not quite as dramatic as this.

The extension of the argument from to is straightforward. The only change is that the orbit now lies in the lattice with density times as large as the density of L. Thus the qualitative predictions of are unchanged, so we predict that Schmidt's conjecture is true for Salem numbers of degree 4 and 6 but not for higher degrees. Quantitatively, the replacement of by . suggests that, 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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