Bijective and General Arithmetic Codings for Pisot Toral Automorphisms.
Let T be an algebraic automorphism of {\Bbb T}^m having the following property: the characteristic polynomial of its matrix is irreducible over \Bbb Q, and a Pisot number  is one of its roots. We define the mapping  &phiv;t acting from the two-sided -compactum onto {\Bbb T}^m as follows: \varphi_t is a fundamental homoclinic point for T, i.e., a point homoclinic to 0 such that the linear span of its orbit is the whole homoclinic group (provided that such a point exists). We call such a mapping an arithmetic coding of T. This paper aimed to show that under some natural hypothesis on  (which is apparently satisfied for all Pisot units) the mapping &phiv;t is bijective a.e. with respect to the Haar measure on the torus. Moreover, we study the case of more general parameters t, not necessarily fundamental, and relate the number of preimages of &phiv;t to certain number-theoretic quantities. We also give several full criteria for T to admit a bijective arithmetic coding and consider some examples of arithmetic codings of Cartan actions. This work continues the study begun in &lsqb;25&rsqb; for the special case m &equals; 2.
Introduction
Let T be an algebraic automorphism of the torus T m given by a matrix M 2 GL(m;Z)
with the following property: the characteristic polynomial for M is irreducible over
Q , and a Pisot number  > 1 is one of its roots (we recall that an algebraic integer
is called a Pisot number if it is greater than 1 and all its Galois conjugates are less
than 1 in modulus). Since det i.e., an invertible element of the
ring We will call such an automorphism a Pisot automorphism. Note
that since none of the eigenvalues of M lies on the unit circle, T is hyperbolic.
This denition is invariant in the following sense: if M is irreducible and one of its
eigenvalues (, say) lies outside the unit disc in the complex plane and all the other
lie inside it, then it is obvious that either  or  is a Pisot number. In the \inverse"
situation (one eigenvalue is inside and the others are outside) it is either  1 or  1 .
We will call T that falls into one of this categories a generalized Pisot automorphism.
Our model will cover all generalized Pisot automorphisms { see Remark 21.
Our goal is to present a symbolic coding of T which, roughly speaking, reveals
not just the structure of T itself but the natural arithmetic of the torus as well. Let
us give more precise denitions.
Let X  denote the two-sided -compactum, i.e., the space of all admissible two-sided
sequences in the alphabet f0; []g. More precisely, a representation of an
of the form
is called the -expansion of x if the \digits" f" k g 1
are obtained by means of the
greedy algorithm (similarly to the decimal expansions), i.e., "
1. The set of all possible sequences
is called the (one-sided) -compactum and denoted by X
.
A sequence whose tail is 0 1 will be called nite.
The -compactum can be described more explicitly. Let
be the
expansion of 1 dened as follows: d 0
2. If the sequence
n g is not nite, we put d n  d 0
n . Otherwise let
k 1), where bar stands for the period of a purely
periodic sequence.
We will write fx n g 1
fy n g 1
fy n g 1
1 and x n < y n for the smallest
such that x n 6= y n . Then by denition,
(see [18]). Similarly, we dene the two-sided -compactum as
Both compacta are naturally endowed with the weak topology, i.e. with the topology
of coordinate-wise convergence, as well as with the natural shifts. Let the -shift
act as follows:
be the corresponding one-sided
shift on X
. For a Pisot  the properties of the -shift are well-studied. Its main
property is that it is soc, i.e., is a factor of a subshift of nite type. In fact this is
equivalent to fd n g 1
being eventually periodic (see, e.g., the review [6]).
We extend the -expansions to the nonnegative integers in the usual way (similarly
to the decimal expansions). There is a natural operation of addition in X  , namely,
if both sequences  " and  " 0 are nite to the left (i.e., there exists N 2 Z such that "
" 00 such that
Later we will show that under some natural assumption on  this operation can be
extended to sequences which are not necessarily nite to the left.
Lemma 1 (see [4], [21]) Any nonnegative element of the ring Z[] has an eventually
periodic -expansion if  is a Pisot number.
Let F in() denote the set of nonnegative x's whose -expansions are nite. Ob-
viously, F in()  Z[] but the inverse inclusion does not hold for an
arbitrary Pisot unit.
Denition 2 A Pisot unit  is called nitary if
A large class of Pisot numbers considered in [11] is known to have this property.
A practical algorithm for checking whether a given Pisot number is nitary was
suggested in [1]. Here is a simple example showing that not every Pisot unit is
nitary. Let r  3, and
i.e.,  is not nitary.
Denition 3 A Pisot unit  is called weakly nitary if for any - > 0 and any
there exists f 2 F in() \ [0; -) such that x
When the present paper was in preparation, the author was told that this condition
had in fact been considered in the recent work by Sh. Akiyama [3], in which
the author shows that the boundary of the natural soc tiling generated by a weakly
nitary Pisot  has Lebesgue measure zero (moreover, these conditions are actually
equivalent). The author is grateful to Sh. Akiyama for drawing his attention to this
paper and for helpful discussions.
A slightly weaker (but possibly equivalent) condition
together with the niteness of fd 0
was used in the Ph.D. Thesis [13] to show that
the spectrum of the Pisot substitutional dynamical system
(where
purely discrete. This claim is a generalization of
the corresponding result for a nitary  from [27] (see also [28]).
Conjecture 4 Any Pisot unit is weakly nitary.
To support this conjecture, we are going to explain how to verify that a particular
Pisot unit is weakly nitary. Firstly, one needs to describe all the elements of the set
has a purely periodic -expansiong: (2)
Lemma 5 (see [3]). The set Z  is nite.
Proof. The sketch of the proof is as follows: basically, the claim will follow from
(see below), which implies that the denominator of any  2 P  in the
standard basis of Q() is uniformly bounded, whence the period of the -expansion
of  is bounded as well.
Therefore, we have a nite collection of numbers f
qg to
\check for periods" (here q is the denominator of  0 dened by (7) in the standard
basis of the ring). Next, it is easy to see that if su-ces to check that Denition 3
holds for any x =  2 Z  (see [3]). Moreover, we can conne ourselves to the case
is the period of . Indeed, if such an f exists,
do as well, and we will be able to make f arbitrarily small. As was shown
in [3], there exists a \universal neutralizing word" f which will suit for all periods of
Z  . Note that all known examples of Pisot units prove to be weakly nitary.
We will need the following technical result.
Lemma 6 A Pisot unit is weakly nitary if and only if the following condition is
there exists such that for any - > 0 and any x 2 Z[]
there exists f 2 F in() \ [-) such that x
Proof. It su-ces to show that if  is weakly nitary, then  in question does
exist. Let  be weakly nitary; then we know that there exists f 0 2 F in() such that
for any  2 Z  we have +f 0 2 F in(). Let  has the -expansion
loss of generality we may regard p to be greater than the
the period of the sequence fd n g 1
(as p is not necessarily the smallest
period of ). Since f 0 can be made arbitrarily small, we may x it such that
for any  2 Z  . Put  := f 0 (provided (3) is satised).
. By Lemma 1, the -expansion of x is eventually periodic, and
splitting it into the pre-periodic and periodic parts, we have
. Let for simplicity of notation whole picture is
shift-invariant). It will now su-ce to check the condition for -
The rst sum in brackets in (4) belongs to F in() and so does the second term. In
view of (3) and the denition of X  , the whole sum in (4) belongs to F in() as well,
because by our choice of p we have necessarily
2 Formulation of the main result and rst steps of
the proof
We recall that the hyperbolicity of T implies that it has the stable and unstable
foliation and consequently the set of homoclinic points. More precisely, a point t 2T m
is called homoclinic to zero or simply homoclinic if T (as is well
known, the convergence to 0 in this case will be at an exponential rate). In other
terms, a homoclinic point t must belong to the intersection of the leaves of the stable
foliation L s and the unstable foliation L u passing through 0. Let H(T ) denote the
set of all homoclinic points for T ; obviously, H(T ) is a group under addition. In [30]
it was shown that every homoclinic point can be obtained by applying the following
procedure: take a point n project it onto L u along L s . Let s denote this
projection; nally, project s onto the torus by taking the fractional parts of all its
coordinates. The correspondence n known to be one-to-one. We will call
-coordinate of a homoclinic point t and n the Z m -coordinate of t.
Note that since T is a Pisot automorphism, we have dimL
We wish to nd an arithmetic coding ' of T in the following sense: we choose
X  as a symbolic compact space and impose the following restrictions on a map
1. ' is continuous and bounded-to-one;
2.
3.
any pair of sequences nite to the left.
In [25] it was shown that if
The proof for an arbitrary m is basically the same, and we will omit it. Our primary
goal is to nd an arithmetic coding that is bijective a.e. Let us make some historic
remarks.
Note that the idea of using homoclinic points to \encode" ergodic toral automorphisms
had been suggested by A. Vershik in [29] for
and was later
developed for a more general context in numerous works { see [30], [16], [24], [25],
[22]. The choice of X  as a \coding space" is special in the case in question; indeed,
the topological entropy of the shift   is known to be log  and so is the entropy
of T . In a more general context (for example, if M has two eigenvalues outside the
unit disc) it is still unclear, which compactum might replace X  . Indeed, since '
is bounded-to-one, the topological entropy of the subshift on this compactum must
have the same topological entropy as T , i.e., log
are the conjugates of , and there is apparently no natural subshift associated with
which has this entropy. However, it is worth noting that the existence of such
compacta in dierent settings has been shown in [30], [15], [22].
Note that if one allows to use slightly altered symbolic transformations (not necessarily
shifts), then there is a hope to give a more explicit expression for the codings
of non-Pisot automorphisms via coding of the higher-rank actions { see examples at
the end of the paper.
Return to our context. The mapping ' t dened by (5) is indeed well dened and
continuous, as the series (5) converges at an exponential rate. Furthermore, since
we have by continuity ' t  does semiconjugate the
shift and a given automorphism T .
We will call ' t a general arithmetic coding for T (parameterized by a homoclinic
point t).
Lemma 7 For any choice of t the mapping ' t is bounded-to-one.
Proof. Let kk denote the distance to the closest integer, s be the R m -coordinate
of t and e
T denote the linear transformation of R m dened by the matrix M . Let 'N;t
be the mapping acting from X  into R m by the formula
s:
Then by (5),
where Therefore, it su-ces to show that
the diameters of the sets N;t (X  ) are uniformly bounded for all N . We have (recall
that


[]


const
is the maximum of the absolute values of the conjugates of  that
do not coincide with . This proves the lemma.
Let the characteristic equation for  be
and T  denote the toral automorphism given by the companion matrix M  for , i.e.,
We rst assume the following conditions to be satised:
1. T is algebraically conjugate to T  , i.e., there exists a matrix C 2 GL(m;Z)
such that
2. A homoclinic point t is fundamental, i.e., hT n t
3.  is weakly nitary.
The notion of fundamental homoclinic point for general actions of expansive group
automorphisms was introduced in [17] (see also [22]).
Remark 8 Note that the second condition implies the rst one, as the mere existence
of a fundamental homoclinic point means that T  T  (see Theorem
Conversely, if T  T  , then there is always a fundamental homoclinic point for T .
Indeed, let n be the Z m -coordinate of t 0 . Then t 0 is a fundamental
for T  and if
Now we are ready to formulate the main theorem of the present paper.
Theorem 9 Provided the above conditions are satised, the mapping ' t dened by
(5) is bijective a.e. with respect to the Haar measure on the torus.
Remark. In [25] the claim of the theorem was shown for 2. We wish to
follow the line of exposition of that paper, though it is worth stressing that our
approach will be completely dierent (rather arithmetic than geometric). In [22]
this claim was shown for any nitary  and it was conjectured that it holds for any
Pisot automorphism satisfying conditions 1 and 2 above. We give further support
for this conjecture, as Theorem 9 implies that we actually reduced it to a general
conjecture veriable for any given Pisot unit  (see Conjecture 4).
The rest of the section as well as the next section will be devoted to the proof of
Theorem 9; in the last section we will discuss the case when conditions 1 and 2 are
not necessarily satised.
We need the following number-theoretic claim. Let
It is obvious that P  is a group under addition.
There exists  Z[] such that
Proof. By the well-known result, for any Pisot ;  2 P  ,  2 Q () and
denotes the trace of an element & of the extension
Q(), i.e., the sum of all its Galois conjugates) { see, e.g., [8]. Since  is a unit,
Thus, if we regard Z[] as a lattice over Z, then by (8), P  is by denition the
dual lattice for Z[]. Hence by the well known ramication theorem (see, e.g., [10,
Chapter III]) the equality (7) follows with
km .
We are going to carry out the proof of the main theorem in several steps.
(description of the homoclinic group).
Lemma 11 Any homoclinic point t for T  has the R m -coordinate
where
Proof. We have M
As was mentioned
above, the dimension of the unstable foliation L u is 1, whence since
Now the claim of the lemma follows from
(7).
Let U  denote the group of units (= invertible elements) of the ring Z[].
Lemma 12 There is a one-to-one correspondence between the group U  and the set
of fundamental homoclinic points for T  . Namely, if t is fundamental, then u in
is a unit and vice versa.
Proof. Suppose t is fundamental. Then the homoclinic point t 0 whose R m -
coordinate is s can be represented as a nite linear integral
combination of the powers T k t, i.e.,
whence u
Therefore, u is invertible in the ring Z[].
Conversely, if u 2 U  , then using the same method, we show that the claim of
the lemma follows from the fact that the equation always has the solution in
Z[], namely,
(reduction to To prove Theorem 9, we may without loss of
generality assume
Then there is a natural one-to-one correspondence between H(T ) and H(T  ), namely,
bijective a.e., then so is 'Ct , as
be a general fundamental homoclinic point for T  given by
(9). In this case the formula (5) turns into the following one:
Step 3 (the pre-image of 0). Let Z  be dened by (2).
Lemma 13 The preimage of 0 can be described as follows:
O  := ' 1
" is purely periodic andX" j  j 2 Z  g:
Proof. By Lemma 7, O  is nite and since it is shift-invariant, it must contain
purely periodic sequences only. Let
whence by (7), u 2 Z[], and  2 Z[], because u 2 U  .
Step 4 (description of the full preimage of any point of the torus). We are
going to show that ' t is \linear" in the sense that for any two sequences  ";
their \dierence" will belong to O  . More precisely, let " (N) denote the sequence
its \value" e (N) :=
. There is an almost
one-to-one correspondence between the set of sequences f" (N) g and R+ , namely
Lemma 14 If ' t any N  1 there exists  2 Z  such that
Proof. Fix
t (fxg) for some x 2 T m and let E denote the set of all partial
limits (in X  ) of the collection of sequences j" (N)
is the sequence
whose \value" is je (N) (e 0 ) (N) j. It
su-ces to show that E  O  . Let
by denition, there exists a sequence
of positive integers fN k g such that - (N k
and we are done.
Therefore, if
(fxg) for some x 2 T m , then we know that to obtain any
t (fxg), one may take one of the partial limits of the sequence f" (N)
for  2 Z  ; perhaps, depending on N . We will write
Conclusion. Thus, we reduced the proof of Theorem 9 to a certain claim about the
two-sided -compactum.
Roughly speaking, our goal now is to show that the procedure described above
will not change an arbitrarily long tail of a generic sequence
will not change  " itself.
3 Final steps of the proof and examples
Let   denote the measure of maximal entropy for the shift (X  ;   ), and
be its
one-sided analog. We wish to prove that
"g.
Step 5 (estimation of the measure of the \bad" set). We will need some basic
facts about the measure   . For technical reasons we prefer to deal with its one-sided
.
Lemma 15 There exists a constant C such that for any n  2 and
any (i 1
,
Proof. Let the mapping
and  n
(C n (")). The Garsia Separation Lemma [12] says that there exists a constant
" and
" 0 are two sequences in X
and
K n . Hence
where L 1 denotes the Lebesgue measure on [0; 1]. Since for any  >
is equivalent
to L 1 and the corresponding density is uniformly bounded away from 0 and 1
(see [19]), we have for some K 0 > 1,
whence by the fact that   is one-to-one except for a countable set of points,
and the claim of the lemma holds with
There is a natural arithmetic structure on X
the sum of two sequences
" and
" 0 is dened as the sequence equal to the -expansion of the sum f
g.
denote the set of nite words of length n that are extendable to a sequence
in
by writing noughts at all places starting with n+1. We will sometimes identify
with the set F in n () 1g.
By the sum ("
In [11]
it was shown that there exists a natural L such that if
F in n () and
().
Recall that by Lemma 6 there exists such that the quantity f
in Denition 3 can be chosen in (-) instead of (0; -). We set
log
and
We can reformulate the hypothesis that  is weakly nitary as follows ( denotes the
-expansion of
for any (" 1
there exists ("
such that (13)
A direct consequence of Lemma 15 is
Corollary 16 For any (i 1
,
Lemma 17 If (i
and (j
, then (i
.
Proof. The claim follows from the denition of X  (see Introduction) and the
fact that the positive root  0 of the equation x is the smallest Pisot number
[9]. Indeed,  5
is a subshift of nite type, namely,
Now the desired claim follows from [18, Lemma 3] asserting that if  0 < , then
Let
We will write
n. The
meaning of the above denitions consists in the fact that if
for some n  1 and by the theorem from [11] mentioned above,
(more precisely, the tail will stay unchanged starting with the (n+L+ 1)'th symbol).
It is obvious that We wish to prove that
1. By the
ergodicity of (X
), we have
su-ces to show that
Proposition There exists a constant
Proof. We have
Y
Since
Y
we have
Now by the formula (14),  being weakly nitary (see (13)) and the denition of L
we have
for any k  1 and any (i
. Hence
and from (16) we nally obtain the estimate
whence one can take
As a by-product we obtain the following claim about the irrational rotations of
the circle by the elements of Z[]. Let, as above,
denote the -expansion of .
Theorem 19 For a weakly nitary Pisot unit  and any  2 Z[] \ [0; 1) we have
-a.e.
.
Proof. We showed that
Conclusion of the proof of Theorem 9. Fix k 2 N . To complete the proof of
Theorem 9, it su-ces to show that the set
has the full measure   . By Proposition 18, for
(D (k)
Hence
(D
(D (k)
and therefore
\
which implies (12). We have thus shown that for   -a.e.
Let L denote the image of   under ' t . Since   is ergodic, so is L and since
log  as well. Hence is the Haar measure
on the torus, as it is the unique ergodic measure of maximal entropy. So, we proved
that
which is the claim of Theorem 9.
As a corollary we obtain the following claim about the arithmetic structure of X
itself.
Proposition 20 Let  denote the equivalence relation on X  dened by (11) and
is a group isomorphic to T m .
Thus, X  is an almost group in the sense that it su-ces to \glue" some k-tuples
(for k < 1) within the set of measure zero to turn the two-sided -compactum for
a weakly nitary Pisot unit  into a group (which will be isomorphic to the torus
of the corresponding dimension). Note that in dimension 2 this factorization can be
described more explicitly { see [25, Section 1].
Remark 21 In fact, we covered all generalized Pisot automorphisms (see the beginning
of Introduction), i.e., Indeed, in the case the same
coding ' will conjugate the inverse shift  1
and T . In the case  the mapping
conjugates  0
and T , where  0
it follows from Proposition 20 that
the operation " 7! " is well dened a.e. on X  .
The following claim is a generalization of Theorem 4 from [26]. Let D(T ) denotes
the centralizer for T , i.e.,
Proposition 22 For a Pisot automorphism whose matrix is algebraically conjugate
to the corresponding companion matrix there is a one-to-one correspondence between
the following sets:
1. the fundamental homoclinic points for T ;
2. the bijective arithmetic codings for
3. the units of the ring Z[];
4. the centralizer for
Proof. We already know that any bijective arithmetic coding is parameterized
by a fundamental homoclinic point. Let t 0 be such a point for T ; then any other
fundamental homoclinic point t satises us 0 , where s 0 and s are the corresponding
R m -coordinates and u 2 U  { the proof is essentially the same as in Lemma 12. On
the other hand, if ' t is a bijective arithmetic coding for T , then as easy to see,
is a toral automorphism commuting with T (this mapping is well dened
almost everywhere on the torus, hence it can be dened everywhere by continuity).
Finally, if u 2 U  and
belongs to GL(m;Z)
and commutes with M and vice versa.
Example 1. (see [24]) Let T be given by the matrix
. Here  is the
golden ratio,
p= 1+2and
Any bijective arithmetic coding for T thus will be of the form
For more two-dimensional examples see [25].
Example 2. Let T be given by the matrix M =@
Here  is the real
root of the \tribonacci" equation x as is well known,  is nitary in
this case (see, e.g., [11]). We have
since Z[] is the
maximal order in the eld Q () and both conjugates of  are complex, again
(recall that by Dirichlet's Theorem, U  must be \one-dimensional", see, e.g., [7]).
Hence any bijective arithmetic coding for T is of the form
22  n 2C AmodZ 3 ;
Example 3. Let M =@ 3 4 1
Here  is the positive root of x
By the result from [2],  is nitary (see Introduction for the denition), and it is easy
to guess that the fundamental units of the ring are  and 1
Besides,
2. Hence any bijective arithmetic coding is either
or
Example 4. Finally, let M =B B @
us show that  is weakly nitary. A direct inspection shows that the only nonzero
period for the positive elements of Z[] is 10000. Hence Z
g. Let, for example,
we have by periodicity
1. The other cases of  2 Z  are similar. Hence
is weakly nitary and we can apply Theorem 9. It su-ces to compute U  ; by
the Dirichlet Theorem, it must be \two-dimensional" and it is easy to guess that the
second fundamental unit (besides  itself) is 1+. Hence U
Zg and the formula for a bijective arithmetic coding can be derived similarly to the
previous examples in view of
4 General arithmetic codings and related algebraic
results
In this section we will present some results for the case when t is not necessarily fundamental
or T is not algebraically conjugate to the companion matrix automorphism.
We will still assume  to be weakly nitary. Let us begin with the case
general t. We recall that there exists a natural isomorphism between the homoclinic
group H(T ) and the group P  , i.e., t be dened as above:
. The question is, what will be the value of #' 1
(x) for a
The next assertion answers this question; it is a generalization of the corresponding
result proven in [25] for and for a nitary  in [23]. Let
the discriminant of  in the eld extension Q()=Q , i.e., the product
are the Galois conjugates of .
Theorem 23 For an a.e. x with respect to the Haar measure,
where N() denotes the norm of an element of the extension Q()=Q .
Proof. Let ' 0 denote the bijective arithmetic coding for T  parameterized by
then one can consider the mapping
A  := '  ' 1
will be well dened on the dense set and we can extend
it to the whole torus. By the linearity of both maps, A  is a toral endomorphism.
Thus, we have
Let A 0
is given by the formula A 0
. For the basis sequence f
with the unity at the rst coordinate we have
Therefore, by the linearity and continuity, we have A
. As ' 0
is 1-to-1 a.e., '  will be K-to-1 a.e. with j. By denition, N(') is the
determinant of the matrix of the multiplication operator x 7! 'x in the standard basis
of Q (), whence given by the companion matrix. Finally,
as by the result from [20, Section 2.7], N( 0
whenever  0 is as in formula (7).
Note that historically the rst attempt to nd an arithmetic coding for a Pisot
automorphism (actually, even for a Pisot endomorphism, i.e., an endomorphism of
a torus with the same property as a Pisot automorphism) was undertaken in [5].
The author considered the case given by the R m -coordinate
From the above theorem follows
Corollary 24 The mapping
is jDj-to-1 a.e.
Suppose now T is not necessarily algebraically conjugate to T  . Let M be, as
usual, the matrix of T , and for n 2 Z m the matrix BM (n) be dened as follows (we
Lemma 25 Any integral square matrix satisfying the relation
Proof. Let B be written column-wise as follows:
(18) and the denition of M  ,
whence by the fact that
Denition 26 The integral m-form of m variables dened by the formula
will be called the form associated with T .
Proposition such that
for Lm -a.e. point x
Proof. Let e
is a certain bijective arithmetic coding for
T  . Then e
B is a linear mapping from T m onto itself dened a.e.; let the same letter
denote the corresponding toral endomorphism. Then e
B. Therefore the matrix B of the endomorphism e
by Lemma and we are done.
As a consequence we obtain
Theorem 28 The minimum of the number of pre-images for an arithmetic coding of
a given automorphism T equals the arithmetic minimum of the associated form f M .
Remark 29 It would be helpful to know whether there is any relationship between
the n in the proposition and the Z m -coordinate of t.
Theorem 30 The following conditions are equivalent:
1. A Pisot automorphism T admits a bijective arithmetic coding.
2. T is algebraically conjugate to T  .
3. The equation
has a solution in
4. There exists a homoclinic point t such that for its Z m -coordinate n,
Proof. (2))(1): see Remark 8;
(1))(2): see the Proposition 27;
also follows from Proposition 27;
(2),(4): it is obvious that M  satises this property (take
Hence so does any M which is conjugate to M  .
Recall that a matrix M 2 GL(m;Z) is called primitive if there is no matrix
2. Following [25], we ask the following
question: can a Pisot toral automorphism given by a non-primitive matrix admit a
bijective arithmetic coding?
Note rst that one can simplify the formula for f M . Namely, since the determinant
of a matrix stays unchanged if we multiply one column by some number and add to
another column, we have
Proposition There exists a sequence of integers N n () such that
More precisely,
a (1)
a (1)
a (1)
(m 1)n
where fa (j)
are dened as the coe-cients of the equation
derived from (6).
Proof. By (19), the denition of a (j)
n and the Hamilton-Cayley Theorem,

a (j)
a (j)
Corollary non-primitive matrix M n 2 GL(m;Z) is algebraically conjugate to
the corresponding companion matrix if and only if so is M , and N n
Let us deduce some corollaries for smaller dimensions.
Corollary 33 (see [25]) For the automorphism given by a non-primitive
admits a bijective arithmetic coding if and only if
Corollary 34 For the matrix algebraically conjugate
to the corresponding companion matrix if and only if  satises one of the
following equations:
1.
2.
3.
Proof. We have N 2
1. The case
thus leads to subcases 1 and 3 and k subcase 2.
Note that if M is the matrix for the \tribonacci automorphism" (see Example 2),
then apparently the only power of M greater than 1 that is algebraically conjugate
to the corresponding companion matrix, is the cube! Indeed, N 2
It seems to be an easy exercise to prove this
rigorously; we leave it to the reader.
Example 5. Let M =@ 1 1 0
and the form
associated with M is (we write
Obviously, the Diophantine equation f M (x; has a solution, namely,
Hence by Theorem 30, M is algebraically conjugate to M  ; for ex-
conjugates them. To show that T admits
a bijective arithmetic coding, it su-ces to check that  is weakly nitary. A direct
inspection shows that the set of periods for the elements of Z  here is f0;
the \universal neutralizing word" (see Introduction) is 13. The author would like to
thank Sh. Akiyama for this computation.
In [25] the author together with A. Vershik considered the case 2. Here if
a b
c d
, then for
and we related the problem of arithmetic codings to the classical theory of binary
quadratic forms. In particular, T admits a bijective arithmetic coding if and only if
the Diophantine equation
is solvable.
The theory of general m-forms of m variables does not seem to be well developed;
nonetheless, we would like to mention a certain algebraic result which looks rele-
vant. Recall that two integral forms are called equivalent if there exists a unimodular
integral change of variables turning one form into another.
Proposition in GL(m;Z) be algebraically conjugate, and
is equivalent either to f M 2
or to f M 2
, and moreover

where A 0 is the transpose of A (we identify a form with the symmetric matrix which
denes it).
Proof. Since M 1 and M 2 are conjugate, they have one and the same characteristic
polynomial. By the denition of f M we have
which is equivalent to (20).
In the end of the paper we would like to present two examples of arithmetic
codings for higher-rank actions on tori that can be obtained as a consequence of the
main construction of this paper. We refer the reader to the recent work [14] for the
necessary denitions and references.
Let the automorphism T of T 3 is given by the matrix from Example 3, namely,
denote the leading eigenvalue. Then the group of units
for Z[] is generated by  itself and (which is also a Pisot unit). Let
T 0 be given by M
it is also a Pisot matrix
algebraically conjugate to the corresponding companion matrix. Then the (Cartan)
action generated by M and M 0 can be encoded by a certain action on X  ; namely,
bijective arithmetic coding of T and  0
given by the formula
It is easy to see that  0
is well dened for almost all the sequences in X  { as  is
will be well dened for any sequence having the block of L zeroes innitely
many times to the left, where L is large enough. Besides, ' conjugates the action
) on X  and the Cartan action (T; T 0 ) on the 3-torus. It is worth noting that
for this example the homoclinic groups for T and T 0 coincide or, in terms of the ring,
One may argue that both generators of this action are both generalized Pisot
automorphisms. This is true but let us give another example, this time of a Cartan
action on T 4 . Namely, let M =B B @
Note M is a companion matrix,
and its spectrum is purely real. Now take the action generated by M
It is easy to check that they all belong to GL(4; Z) and that
this will yield a Cartan action on T 4 as well as the fact that the dominant eigenvalue
of M is indeed weakly nitary. We leave the details to the reader. Therefore,
the usual encoding mapping ' conjugates the action (  ;    id) on the
compactum X  and the Cartan action has two eigenvalues
inside the unit disc and two outside. Perhaps, this is the rst ever explicit bijective
a.e. encoding of a non-generalized Pisot automorphism (though not by means of a
shift).
We believe that the underlying ideas of these examples can be extended to more
general Cartan actions; however, there are obvious problems that arise in doing so.
They are as follows:
1. to show that any Pisot unit with the real conjugates is weakly nitary (see

Introduction

2. to nd out whether a given Cartan action contains a Pisot automorphism whose
matrix is conjugate to its companion matrix (of course, every Cartan action is
known to contain some Pisot automorphism);
3. if so, nd it in such a way that a given element of  is a linear integral
combination of the powers of this Pisot automorphism. For instance, this is
true if the ring Z[] is the maximal order of the eld Q (), where  is, as usual,
the dominant eigenvalue of the matrix of the Pisot automorphism in question
{ see [14].
We plan to develop this direction elsewhere.

Acknowledgement

. This research was supported by the EPSRC grant number
GR/L98923. The author wishes to thank Anatoly Vershik for our fruitful collaboration
in dimension 2 in [25] and numerous helpful suggestions. The author is grateful
to A. Manning for stimulating discussions that led to the previous example.



--R

Number theory (Eger
Cubic Pisot units with
On the boundary of self-a-ne tiling generated by



Number Theory
An Introduction in Diophantine Approximation
Dufresnoy and C.

Theory Dynam.
Arithmetic properties of Bernoulli convolutions
Linear Numeration Systems
Rigidity of measurable structure for Z d - actions by automorphisms of a torus
Arithmetic construction of so
Un codage so
Homoclinic points of algebraic Z d


Algebraic Theory of Numbers
On periodic expansions of
Algebraic codings of expansive group automorphisms and two-sided beta-shifts
An arithmetic group associated with a Pisot unit
Ergodic properties of Erd
Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus
Bijective codings of automorphisms of the torus

The simultaneous action of adic transformation and Markov shift on torus

Arithmetic isomorphism of the toral hyperbolic automorphisms and so
--TR
pisot automorphism;integral form;arithmetic coding;homoclinic point