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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

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

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

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

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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