Download , ,n N X N X

1106
Communications of the Moscow Mathematical Society
Bijective coding of automorphisms of
the torus and binary quadratic forms
A. M. Vershik and N. A. Sidorov
We consider special maps from a symbolic compactum to a two-dimensional torus that connect
a bidirectional translation with a given hyperbolic automorphism of a two-dimensional torus and
preserve the maximum entropy measure. In contrast to the known geometric coding method with
the aid of Markovian partitions (see references in [2] and [9]), the coding under consideration uses
the arithmetic and group theoretic structure. This makes it possible to connect the arithmetic of
the corresponding algebraic extension of Q and the properties of a certain binary quadratic form
with the dynamics of the automorphism. The transformations can be parametrized by the units
of the quadratic field or by the homoclinic points of some specific orbit.
Let T be a hyperbolic automorphism of T2 given by the matrix M = ac db ∈ GL(2, Z). We put
r = Tr M, σ = det M, D = r 2 − 4σ. Hyperbolicity implies that r√6= 0 for σ = −1, and |r| ≥ 3 for
σ = +1. For simplicity, we shall assume that r > 0. Let λ = (r + D)/2 be the greatest eigenvalue
of M . As a symbolic compactum X we choose either a stationary Markov compactum Xr with
state space 0, 1, . . . , r and pairwise prohibitions {εn = r ⇒ εn+1 = 0, n ∈ Z} for σ = −1, or a sofic
s−2(r −1) for any n ∈ Z
compactum Yr = {{εn }∞
−∞ : 0 ≤ εn ≤ r −1, (εn . . . εn+s ) 6= (r −1)(r −2)
and any s ≥ 2} for σ = +1. Therefore, each of the two compacta is fully determined by the trace
and is a β-compactum for β = λ (see [7]). Let τ denote a bidirectional translation on X, that is,
(τ ε)n = εn+1 , and let µ be a measure of maximum entropy. We define the normalization operation
Q
l0
as a map n : ∞
−∞ {0, 1, . . . , 2r} → X such that if {xk } is a finite sequence, then n({xk }) = {εk }l
is the sequence of coefficients of a Markovian (sofic) β-decomposition with β = λ of the number
P
x = k xk λ−k . More precisely, l = [− logλ x] + 1, εl = [λl x], εk = [λS k−l(λl x)], k > l, where
Sx = {λx}. For an arbitrary sequence the normalization is defined as the weak limit of normalizations of finite approximations. On the well-posedness of this definition, see [4] and [9]. The sum
of two sequences in X is defined to be the normalization of their coordinatewise sum. Similarly,
one can define the inverse element with respect to addition. It can be shown that the sum and
difference are defined for a.a. pairs of sequences under the measure µ × µ. After the natural arithmetic factorization mod 0, the maximum entropy measure p : X → X0 defines a group structure
on the quotient space X0 , with respect to which the translation τ is a group automorphism.
We recall that a point s ∈ T2 is said to be homoclinic to zero (briefly: homoclinic ) if it belongs
to the intersection of the leaves of the expanding and the contracting foliations passing through 0.
Each homoclinic point s can be obtained in a unique way as the projection of a lattice point
z(s) ∈ Z2 onto the one-dimensional eigenspace Lu corresponding to the eigenvalue λ along the
other one-dimensional eigenspace Ls , followed by projection of the resulting point t = t(s) onto
the torus (see [1]). The point t = t(s) will be called the plane coordinate of the homoclinic point s.
Lemma 1. The plane
of any
homoclinic point of T ,has the following
form: t(s) =
√ coordinate √
,
,u
n
t(u, v) = (n + uλ)/ D, (k + vλ)/ D , where (u, v) = z(s) and k = −σM −1 v .
Definition. The composite map X → X0 → T2, where the first map is a factorization of p (see
above) and the second is a group homomorphism X0 → T2 taking τ into T , is called an arithmetic
coding of the automorphism T .
Theorem 2.
ϕs (ε) =
Any arithmetic coding of a hyperbolic automorphism T has the form
lim
N →+∞
X
N
−N
εn T −n t(s)
mod Z2 =
lim
N →+∞
X
N
−N
εn λ−n t(s) mod Z2 ,
(1)
where s is a non-zero homoclinic point of T , t(s) is the plane coordinate of s, and multiplication
under the summation sign is understood as the multiplication of a number by a vector in R2.
Communications of the Moscow Mathematical Society
1107
Definition. The binary quadratic form fT (x, y) = cx2 + (a − d)xy − by2 is called the form
associated with T .
Theorem 3. The map ϕs ( · ) transforms µ into the Haar measure on the torus. The number of
inverse images of almost every point is equal to K , where K = |fT (u, v)| and (u, v) = z(s).
If ϕs is a bijection a.e., we shall call such a coding a bijective arithmetic coding (BAC). It
turns out that for many automorphisms T a BAC does not exist.
Theorem 4. 1. The following assertions are equivalent.
(a)
The hyperbolic automorphism T given by the matrix M admits a bijective arithmetic
coding.
r 1
in GL(2, Z) to the associate matrix Cr,σ = −σ
, where
0
(b) M is algebraically adjoint
r = Tr M and σ = det M .
(c) The Diophantine equation
fT (x, y) = ±1
(2)
is soluble.
(d) There is a point (x, y) ∈ Z2 such that the linear span of the orbit of this point under the
action of the powers of M is Z2. Any such√point satises equation (2).
2. Suppose that T admits a BAC and λ 6= 3+2 5 . There exists a unique one-to-one correspondence between the set of bijective arithmetic codings T , the Dirichlet group (T ) =
{A : AT = T A} =√{±T k | k ∈ Z} and the group of units U(D) = {±λk | k ∈ Z} of the
quadratic eld Q( D). Namely, if ϕs1 , ϕs2 are two BAC's, then ϕs2 = Aϕs1 for some
A ∈ U(T ), and t(s2) = δt(s1 ), where δ ∈ U(D).
In the case when M is not adjoint to the associate matrix, a minimal arithmetic coding serves
as an analogue of a BAC. This is defined as a coding with the smallest possible number m of
inverse images. Theorem 3 implies that m = min {|fT (u, v)| : (u, v) ∈ Z2 \ (0, 0)}, that is, m is
the arithmetic minimum of the quadratic form fT .
The ideas of symbolic coding of automorphisms of the torus with the aid of arithmetic decompositions appeared in [1], have been considered from a different viewpoint in [3], and have been
discussed and developed in [5], [6], [8]–[10].
Some of these results can be generalized to hyperbolic automorphisms of many-dimensional
tori. This case will be considered in detail in forthcoming papers.
The complete version of the proofs of the assertions in this article can be found in [9].
Bibliography
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Funktsional. Anal. i Prillozhen. 26: 3 (1992), 22–24; English transl.,
Functional Anal. Appl. 26: 3 (1992), 170–173.
R. L. Adler, Bull. Amer. Math. Soc. 35 (1998), 1–56.
A. Bertrand-Mathis, Bull. Soc. Math. France 114 (1986), 271–323.
C. Frougny and J. Sakarovitch, Internat. J. Algebra Comput. (to appear)
R. Kenyon and A. Vershik, Ergodic Theory Dynamical Systems 18 (1998), 357–372.
S. Le Borgne, Dynamique symbolique et propri
etes stochastiques des automorphisms du tore:
cas hyperbolique et quasi-hyperbolique, Thèse doctorale, 1997.
W. Parry, Acta Math. Hungar. 11 (1960), 401–416.
A. M. Vershik,
N. A. Sidorov and A. M. Vershik, “Ergodic properties of Erdös measure, the entropy of the
goldenshift, and related problems”, Preprint IMS 14/96; to appear in Monatsh. Math.
[9] N. A. Sidorov and A. M. Vershik, “Bijective arithmetic codings of hyperbolic automorphisms
of the 2-torus, and binary quadratic forms”, Preprint IML 98/21; to appear in J. Dynam.
Control Systems.
[10] M. Einsiedler and K. Schmidt, Trudy Mat. Inst. Steklov. 216 (1997), 265–284; English
transl. in Proc. Steklov Inst. Math. 216 (1997).
Steklov Mathematics Institute
Russian Academy of Sciences
St. Petersburg Division
[email protected], [email protected]
Received 19/AUG/98