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The beta transformation was introduced by Rényi [6] as an example of a simple dynamical system. Given , the

If is not an integer, we can use **T** to set up an expansion for
which the above description is still valid. Define and
. Then ,
for . Clearly, for all **n**, and

Here is some code to compute the beta expansion of alpha (and evaluate the above sum).

One distinction between integer and non-integer bases is that, in the latter case,
not every possible sequence is a possible expansion.
Parry [5] gave a simple criterion for which sequences can occur:
Let be the modified beta expansion of **1** defined as above, except that the
fractional part is taken to be in rather than . This prevents the
expansion from ending in an infinite sequence of zeros. Then is the expansion
of some in if and only if dominates in the
sense of lexicographic order. Another point is that if is not an integer
there can be infinitely many expressions of as a sum (1.3) with all
satisfying . The expansion we have singled out is the
``greedy'' expansion.

You can verify this lack of uniqueness for yourself in case
is the golden ratio. Notice that the relation
enables one to replace a word **100** in such an expansion
by **011**. So clearly some numbers will have uncountably many expansions to this
base, but only one of these is * the* beta expansion.

Recall the familiar argument which shows the periodicity of the **b** expansion of
a rational number . It is clear that all the numbers
are of the form with . Since there are only a finite
number of possibilities for , there must be a repetition in the sequence
and hence in the sequence . But determines uniquely,
and hence is periodic and thus is periodic.

Let us consider whether there could be non-integers with the same property. It is easy to see by induction that , where

Thus, if the expansion of is periodic, so that , with the period lengthHowever, it does not seem likely at first glance that would be periodic if is a non-rational algebraic integer. It is clear that all of the numbers are in the ring , i.e. of the form

where
Recall that a
Pisot number
(or * Pisot-Vijayaraghavan number*, or * PV
number*) is an algebraic integer for which all conjugates of
with satisfy . (The conjugates are
the roots of the minimal polynomial). For
example, the golden ratio is a Pisot number since its conjugate satisfies
. There are Pisot numbers of every degree,
in fact there are such numbers in every number field.

Bertrand [1] and Schmidt [7] independently showed that if is rational and is a Pisot number then is periodic. It makes sense to test this somewhat unexpected result with a few experiments. For example, take and . This is easy to try numerically on a pocket calculator, or using Maple, and we find the values for for shown in Table 1. This certainly suggests that the expansion is purely periodic with period 8. We can verify this by carrying out the calculation in the ring , or let Maple do it and obtain the exact values for shown in Table 2.

Why is this periodic? The result becomes less mysterious if we realize that the numbers listed in Table 2 are the projections onto the real axis of points in the lattice , i.e. the set of integer combinations of and . Since is irrational, the set of such projections is dense in , so why does the above orbit only meet finitely many of these lattice points? This is where the condition that be a Pisot number is used. Notice that the conjugate of is . But since and all , we have

So, we are only dealing with a finite portion of the lattice , namely that contained in the rectangle , and this is only a finite set of lattice points.

The following code enables you to change the values of beta and alpha.

It is clear how this argument extends to any rational and any Pisot number .
The only difference is that if has degree **d**, we are dealing with a lattice in
. The bounds one obtains on the size of the orbit are quite large and do not
suggest the many patterns one observes in practice. For a discussion of the
expansions for Pisot numbers in , and an answer to the question of whether
the
complementary factor
need be cyclotomic, see
[3].

As mentioned above, Schmidt showed that if the -expansion of every rational is
periodic then must be a Pisot or a
Salem number
. Recall here that a
*
Salem number
* is an algebraic integer such that all conjugates satisfy
with at least one case of equality. This implies that is
reciprocal, so is a conjugate of and the minimal polynomial of
is a reciprocal (palendromic) polynomial of even degree.
Schmidt's conjecture is that the converse of this theorem is true, that is that
if is a Salem number, then the -expansion of every rational is periodic.

The smallest degree for Salem numbers is 4.
The behaviour of the preperiod and period lengths is quite tame in this case,
as one can see from
Table 3.
, which gives the period and preperiod for each
Salem number of degree 4 and trace 10. Each such number is identified by
the middle coefficient of the minimal polynomial, i.e. satisfies
.

Click here
to compute the first few terms of the Beta Expansion of a degree 4
Salem number from its minimal polynomial .

The pattern observed in
Table 3.
is general, as we showed in [2], where we
proved Schmidt's conjecture for and all Salem numbers of degree 4
(so is a beta number in Parry's terminology).
The general conjecture remains open even for degree 4. The method used in
[2] could likely be extended to other rationals with small denominators
but it is unlikely to succeed for all rational .
Explicitly, if , then in all cases **m = 1**, i.e., (mod **1**) is a purely
periodic point of **T**. If has the minimal polynomial , then for fixed **a** the period is a unimodal function of **b** and
takes on values which lie in the set even .
Thus, for all Salem numbers of degree **4**.
No such bound seems to be true for Salem numbers of higher degree, even for .

In contrast, for degree 6, we find a much more interesting behaviour, as one sees
by inspecting the
tables
in [4].
Table 4
reproduces some of that data
for the Salem numbers with minimal polynomials
. In all cases but one, **m = 1** and .
But for **c = -7**, we have only been able to show that **m + p > 1199978517**, so the orbit
in this case is very large, possibly infinite. Clearly **c = -7** is notable not only for
the size of its orbit but also the size of . Our discussion below will
indicate why a small value of should be expected to lead to a large value
of **m + p**. More precisely, we suggest a relation between the size of the orbit and
the size of .

Click here
to compute the first few terms of the Beta Expansion of a degree 6
Salem number from its minimal polynomial .

At first it is natural to suspect that in this case the orbit is infinite. In order to
understand why we now feel this is not the case, we must consider the probabilistic
model presented below and, in more detail, in [4]. What we believe is true
here is that **m** and **p** are simply very large. Such large periods are possible.
For example, for the Salem number with minimal polynomial
, the preperiod and period of the
orbit of **1** are **m = 39420662** and **p = 93218808**. For other large values of **m+p**, see
Table 2
of [4].

There are **11836** Salem numbers of degree **6** and trace at most **15**. We have computed
the complete expansion for all but **80** of these. For the most part there is a great
deal of regularity in the distribution of , and the values of **m** and **p** are quite
small: for all but **199** of the numbers surveyed, both **m < 1000** and **p < 1000**. For
9609 of the numbers (81% of the total) we have **m = 1**, but, in contrast to the
degree-**4** situation, larger values of **m** do occur with a certain regularity. However,
among the remaining **199** cases, there are at least **79** for which and
at least two for which .

Our probabilistic model suggests that **m + p** will be large if is small
relative to .
A competing explanation might be that some peculiar arithmetic relation between the
conjugates of on the unit circle is the cause of the large orbits. To see that
this is not the case, consider the orbits of **1** when is a power of the smallest
Salem number of degree 6, which has minimal polynomial . This data is given in
Table 5.
. The correlation between the large
values of **m+p** and the large values of
is evident.

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