Quasi-linear transformations, substitutions,
numeration systems and fractals
Marie-Andrée Jacob-Da Col and Pierre Tellier
LSIIT-UMR 7005, Pôle API, Bd Sébastien Brant
67412 Illkirch Cedex France
[email protected], [email protected]
Abstract. In this paper we will define relations between quasi-linear
transformations, substitutions, numeration systems and fractals. A QuasiLinear Transformation (QLT) is a transformation on Zn which corresponds to the composition of a linear transformation with an integer
part function. We will first give some theoretical results about QLTs. We
will then point out relations between QLTs, substitutions, numerations
systems and fractals. These relations allow us to define substitutions,
new numeration systems and fractals associated with them. With help
of some properties of the QLTs we can give the fractal dimension of these
fractals.
Keywords: Gaussian integers, numeration systems, discrete linear transformations, substitutions, fractals
1
Introduction.
Fractal tiles generated by numeration systems and substitutions has been widely
studyed, see for example [1],[2], [5]. In this paper we define discrete linear transformations called QLT which generate tilings of Zn . The tilings studied here are
self-similar, they allow us to generate n-dimensional fractals. In dimension two
we define substitutions that generate the border of the tiles, and we point out
relations between QLTs and numeration systems which allow us to define new
numeration systems. These discrete linear transformations are also in relation
with discrete lines [17],[].
Let g be a linear transformation from Zn to Qn , defined by a matrix w1 A
where A is an integer matrix and w a positive integer. If we compose this transformation with an integer part function we obtain a Quasi-Linear Transformation
(QLT) from Zn to Zn . We will only consider the integer part function with a
positive remainder. We will denote it ⌊ ⌋. If x and y are two integers, ⌊ xy ⌋ denotes
the quotient of the Euclidean division of x by y. We denote G as the QLT defined
by g. Main research results on such transformations are listed in the first section
of this paper.
A substitution on an alphabet A is an application that associates each letter
of A with a word over the alphabet A. In the second section of this paper we
consider sequences of tiles associated with QLTs of Z2 . We define substitutions
2
Marie-Andrée Jacob-Da Col and Pierre Tellier
that generate the border of these tiles. Each sequence of tiles converges (after
renormalization) toward a tile wih a fractal border whose fractal dimension is
computed with help of the substition.
Let us consider β an algebraic number and D a set of elements of Z[β]. β
is a valid base using the digits set D if every integer c ∈ Z[β] has a unique
n
P
aj β j , where aj ∈ D and
representation (decomposition) of the form: c =
j=0
n ∈ N. Then (β, D) is also called a numeration system. The third section will
point out some relations between QLTs and numeration systems when β is a
Gaussian integer (β = a + ıb, with a, b ∈ Z) or an algebraic integer of order 2
(β 2 + bβ + a = 0 with a, b ∈ Z). These relations and some properties of QLTs
will allow us to generate new numeration systems.
In the fourth section we will see how QLTs generate fractals (some of them are
in relation with substitutions and numeration systems). Their fractal dimension
can be determined by the substitution rules associated with the QLT or directely
using some properties of the QLT.
2
Quasi-linear transformations
In this section we recall definitions and results that can be found in [7], [10], [3],
[4] and that are useful in the rest of the paper.
Definition 1 Let g be a linear transformation from Zn to Qn , defined by a matrix w1 A where A = (ai,j )1≤i,j≤n is an integer matrix and w a positive integer.
The Quasi-Linear Transformation or QLT associated with g is the transformation of the discrete plane Zn defined by the composition of g with the greatest
integer part function noted ⌊ ⌋. The QLT is then noted G.
In the following we will denote δ = det(A), where det(A) is the determinant
of A, and we’ll assume that δ > 0. The linear transformation defined by the
matrix ω1 A extends to Rn and for each point Y of IRn , there exists a unique
point X of IRn such that Y = g(X). This is not the same for a QLT. Indeed,
each point of Zn can have either none, one or several antecedents: the set of
antecedents of X ∈ Zn is then called tile with index X.
Definition 2 We call tile with index X ∈ Zn and denote PG,X the set:
PG,X = {Y ∈ Zn |G(Y ) = X}
Definition 3 We call p-tile or tile of order p ∈ N, with index X ∈ Zn , and
p
p
denote PX
the set: PG,X
= {Y ∈ Zn |Gp (Y ) = X} where Gp denotes the transformation G iterated p times.
Definition 4 Two tiles are geometrically identical if one is the image of the
other by a translation of an integer vector.
Quasi-linear transformations, substitutions, numeration systems and fractals
3
In this paper we focus on a particular type of QLT, called ”m-determinantal
QLT”.
Definition 5 A QLT defined by a matrix w1 A such that w = m det(A) where m
is a positive integer, is called a m-determinantal QLT. A 1-determinantal QLT
will be called a determinantal QLT.
In the following G will always denote a m-determinantal QLT associated
1
A, and g the associated rational linear transformation. To
with the matrix mδ
p
p
simplify the notations, tiles PG,X and PG,X
will be noted PX and PX
. Moreover
(0, 0, . . . , 0) ∈ Zn will be simply noted O.
Proposition 1. The tiles generated by a m-determinantal QLT G are all geometrically identical. For all Y ∈ Zn , PY = TmAbT Y PO where Tv represents the
bT is the transpose of the cofactor matrix of A.
translation of vector v and A
bT is the transpose the cofactor matrix of A, we have A−1 = 1 A
bT .
Proof. : If A
δ
bT Y + X. We then have:
Let us consider
Y ∈ Zn , X ∈ iPO and X ′ = mA
j
1
1
bT Y = 1 AX + Y = G(X) + Y . It follows that
G(X ′ ) = mδ
AX + mδ
AmA
mδ
bT Y + X is a point of PY . We can now conclude
for each point X ∈ PO , X ′ = mA
that PY = TmAbT Y PO .
⊓
⊔
Proposition 2. The p-tiles generated by a m-determinantal QLT are all geometrically identical and can be generated recursively by translations of PO . More
precisely, if Tv refers to the translation of the vector v we have, for all p ≥ 1:
and
PYp = T(mAbT )p Y POp
[
[
POp+1 =
T(mAbT )p X POp =
T(mAbT )X PO
(1)
(2)
p
X∈PO
X∈PO
Proof. : If p = 1, the equality (1) has been proved in proposition 1.
By definition PO2 = {X ∈ Zn |X ∈ PY ′ and Y′ ∈ PO } so
PO2 =
[
Y ′ ∈PO
PY ′ =
[
Y ′ ∈PO
TAbT
Y ′ PO
which proves the equality (2) for p = 1.
Let us now assume the equalities (1) and (2) to be true with p = k and prove
that they are true for p = k + 1.
4
Marie-Andrée Jacob-Da Col and Pierre Tellier
PYk+1 =
=
S
X∈P
SY
k
PX
X ′S
∈PO
k
PX
′ +mA
bT Y
T mAbT k (X ′ +mAbT Y ) POk
(
)
S
T mAbT k X ′ POk
= T mAbT k+1 Y
)
)
(
(
X ′ ∈PO
=
(recurrence hypothesis)
X ′ ∈PO
Moreover :
= T mAbT k+1 Y POk+1
)
(
POk+1 =
=
S
X∈P
SO
X∈PO
And :
POk+1 =
=
S
(recurrence hypothesis)
k
PX
T mAbT k X POk
)
(
(recurrence hypothesis)
PX
k
X∈PO
S
k
X∈PO
T(mAbT )X PO .
⊓
⊔
Figure 2 shows the tile of order 2 of the QLT defined by
For example,
−3 5
, this tile is composed of 34 tiles of order 1. The following proposi−5 −3
tion determines the number of points of a tile of order n and will be used in the
last section to detemine the fractal dimension of n-dimensional fractals.
1
34
Proposition 3. The number of points of a p-tile generated by a m-determinantal
QLT in Zn is equal to δ p(n−1) mnp .
Proof. Let first proof that the tile PO contains exactely δ n−1 mn points.
– Case of m = 1 and n = 2: it is know (see []) that in this case the number of
points of P0 equals δ.
– Case of m = 1 and n > 2. It has been proved in [] and [] that for each integer
matrix A it exists an unimodular matrix U such that AU = B where B is
an upper triangular integer matrix and that the point of the tiles of A are in
one-to-one correspondance with the points of the tiles f B. Therefore we will
only do the proof for an upper triangular integer matrix B = (bi,j )1≤i,j≤n .
′
Let denote P0 , the tile associated with B and PX
the tiles associated with
′
′
′
′
B = (bi,j )1≤i,j≤n−1 with bi,j = bi+1,j+1 and δ = b2,2 b3,3 . . . bn,n . Suppose
Quasi-linear transformations, substitutions, numeration systems and fractals
5
′
that the number of points PX
equals δ ′n−2 . We have:
(x1 , x2 , . . . , xn ) ∈ PO
0 ≤ b1,1 x1 + b1,2 x2 + . . . + b1,n xn < δ
0≤
b2,2 x2 + . . . + b2,n xn < δ
⇔ .
.. ..
..
. .
0≤
bn,n xn < δ
b x +...+b1,n−1xn−1 +b1,n xn
≤
− 1,2 2
b1,1
⇔
⇔
x1 < δ ′ −
0≤
..
.
b2,2 x2 +...+b2,n xn
δ′
0≤
bn,n xn
δ′
b x +...+b1,n−1xn−1 +b1,n xn
− 1,2 2
b1,1
(x2 , x3 , . . . , xn ) ∈ Pi′2 ,i3 ,...,in
b1,2 x2 +...+b1,n−1xn−1 +b1,n xn
b1,1
<
b1,1
..
.
b1,1
..
.
≤
x1 < δ ′ −
with ik = 0, 1, . . . , b1,1
<
b1,2 x2 +...+b1,n−1xn−1 +b1,n xn
b1,1
The number of solutions for x1 equals δ ′ and each tile Pi′2 ,i3 ,...,in contains
n−1
δ ′n−2 , therefore we have δ ′ δ ′n−2 b1,1
= δ n−1 points.
– If m > 1, we have:
(x1 , x2 , . . . , xn ) ∈ PO
0 ≤ a1,1 x1 + a1,2 x2 + . . . + a1,n xn < mδ
..
..
⇔ ...
.
.
0 ≤ an,1 x1 + an,2 x2 + . . . + an,n xn < mδ
a x +a x +...+a1,n xn
<m
0 ≤ 1,1 1 1,2 δ2
⇔
a2,1 x1 +a2,2 x2 +...+an,n xn
0≤
<m
δ
⇔ (x1 , x2 , . . . , xn ) ∈ Pi′1 ,i2 ,...,in with i1 , i2 , . . . , in = 0, 1, . . . , m − 1
′
But each Pi,j
contains δ n−1 points, it follows thats PO contains δ n−1 mn
points.
We conclude that the number of tiles of PO equals δ n−1 mn .
Let now supose that the number of points of POp equals δ p(n−1) mnp , we have
S
POp+1 = X∈PO T(mAbT )p X POp (see proposition 2), it follows that the number of
points of POp+1 equals δ p(n−1) mnp δ n−1 mn = δ (p+1)(n−1) mn(p+1) .
⊓
⊔
It is well known that if g is a contracting linear transformation of Rn then
g has the origin as unique fixed point and for each X ∈ Rn the sequence g n (X)
tends toward this fixed point. We will say that a QLT G is contracting if g
is contracting. But such a QLT has not necessarily a unique fixed point. The
behavior under iteration of 2D contracting QLTs has been studied in [7], [16],
[14] and [15]. The following definition and theorem will be used to define new
numeration system.
Definition 6 A consistent Quasi-Linear Transformation is a QLT which has
the origin as unique fixed point such that for each discrete point Y the sequence
(Gn (Y ))n≥0 tends toward this unique fixed point.
6
Marie-Andrée Jacob-Da Col and Pierre Tellier
Consider a 2D-QLT defined by
||g||∞ =
1
ω
ab
, the infinite norm of g is then
cd
1
max(| a | + | b |, | c | + | d |).
ω
The two following theorems, proved in [7], states conditions such that a 2DQLT is a consistent QLT.
Theorem 1. Let G be a QLT such that ||g||∞ < 1, G is a consistent QLT if
and only if one of the three following conditions is verified:
(1) b ≤ 0, a + b ≤ 0, c > 0 and
d≤0
(2) a ≤ 0,
b > 0, c ≤ 0 and c + d ≤ 0
(3) a ≤ 0,
b ≤ 0, c ≤ 0 and
d≤0
Theorem 2. Let G be a QLT such that ||g||∞ = 1, G is a consistent QLT if
and only if one of the following conditions is verified:
1. If |a| + |b| = ω and |c| + |d| < ω
– a = 0, b < 0, c > 0 and c − d ≥ 0
– a > 0, a + b ≤ 0, c > 0 and d ≤ 0
– a ≤ 0, b > 0, c ≤ 0 and c + d ≤ 0
– a < 0, b < 0, c ≥ 0 and d ≤ 0
2. If |a| + |b| < ω and |c| + |d| = ω
– d = 0, c < 0, b > 0 and b − a ≥ 0
– d > 0, c + d ≤ 0, b > 0 and a ≤ 0
– d ≤ 0, c > 0, b ≤ 0 and a + b ≤ 0
– d < 0, c < 0, b ≥ 0 and a ≤ 0
3. If |a| + |b| = ω and |c| + |d| = ω
– ω = −a + b = −c ± d, a + b ≤ 0, c + d ≤ 0 and abc 6= 0
– ω = −b = c + d, c − g ≥ 0, and bcd 6= 0
– ω = a − b = c − d, a + b ≤ 0, c + d > 0 and bcd 6= 0
– ω = b = −c ± d, c + d ≤ 0 and bcd 6= 0
– ω = −a − b = c − d, a − b < 0, c + d > 0 and abcd 6= 0
– ω = −a ± b = c − d, a + b ≤ 0, c + d ≤ 0 and bcd 6= 0
– ω = a + b = −c, a − b ≤ 0 and abc 6= 0
– ω = a − b = −c + d, a + b > 0, c + d ≤ 0 and abc 6= 0
– ω = ±a − b = c, a + b ≤ 0 and abc 6= 0
– ω = −a + b = −c − d, a + b > 0, c − d > 0 and abcd 6= 0
3
Quasi-linear transformations and substitutions
In this section we only consider 2D m-determinantal QLTs.
A substitution on an alphabet A is an application that associates each letter
of A with a word over the alphabet A. In this section we will define a substitution
over an alphabet A of 16 letters. The words generated by these substitutions
Quasi-linear transformations, substitutions, numeration systems and fractals
7
corresponds to the Freeman code of the border of the tiles. The substitutions
will be used to determine the fractal dimension of the border.
Let us remember that a discrete line is the set of points
ax + by
2
=i
D(a, b, ω, i) = (x, y) ∈ Z |
ω
where a, b, i, ω ∈ Z and ω > 0 (see [17]).
A discrete line such that ω = max(| a |, | b |) is called a discrete naive line
and will be noted N (a, b, i). The geometrical structure of a naive line is periodic,
we will use this periodicity to determine the border of a tile.
The tile with index (i, j) is the intersection of the two discrete lines D(a, b, mδ, i)
and D(c, d, mδ, j) . These discrete lines have been studied (see [17]) for coprime
coefficients, so we note t = gcd(a, b), t′ = gcd(c, d), a′ = a/t, b′ = b/t, c′ = c/t′
and d′ = d/t′ . The study of discrete lines [17] and QLTs [15],[16], [7], [9] induce
the following properties.
Property 1. 1. PO is geometrically identical to Pi,j , ∀i, j ∈ Z;
2. PO contains m2 δ points;
3. PO contains exactly m.t translates of the period of the naive line N (a′ , b′ , 0);
4. PO contains exactly m.t′ translates of the period of the naive line N (c′ , d′ , 0);
5. PO has exactly 6 neighbours (or 4 neighbours if a = d = 0 or b = c = 0)
among the sets Pi,j with i, j ∈ {0, ±1}.
−3 5
1
,
Figure 1 corresponds to the tile PO associated to the QLT of matrix 34
−5 −3
we see also the period of the naive lines defined by N (−3, 5, 0) and N (−5, −3, 0).
The border of PO can be split into six parts which are the frontiers between PO
and its six neighbours. These six parts are noted Γi with i = 0, 1, ..., 5 as shown
in figure 1. The arrows indicate the direction. With each border is associated a
word whose letters correspond to the direction of the successive segments (U =
up, D = down, L = left, R = right). In our example: Γ0 =RURRURR, Γ1 =U,
Γ2 =LULUULU, Γ3 =LLDLLDL, Γ4 =D and Γ5 =DRDDRD.
We now explain how to obtain Γi for i = 0, 1, ..., 5. We limit the study to the
case a′ < 0, b′ > 0, | b′ | ≥ | a′ |, c′ < 0, d′ < 0 and | c′ | ≤ | d′ |; the other cases
are similar and can be obtained by symmetry.
′
a x + b′ y
– Γ0 is the bottom of mt times the period of the naive line
= 0.
b′
Réveilles [17] proved that the period of a naive line contains
and short
long
b′
. The steps
steps. We will note them l and s. We will also note q =
−a′
s and l have the respective length q and q + 1. Let [0, q1 , q2 , . . . , qn ] be
−a′
the continued fraction development of ′ . The period of the naive line
b
′
a x + b′ y
= 0 can be obtained by the following recurrence (Réveilles [17]):
b′
µ0 = l
8
Marie-Andrée Jacob-Da Col and Pierre Tellier
sequence LL
points (-1,1), (-2,1) and (-2,2)
sequence LD
belong to PO ; the point (-1,2)
doesn’t belong to PO
so we have the sequence LU
in the word of the border
(0,0)
(-1,0)
P−1,1 , P−2,1 and P−2,2
2; P
belong to PO
−1,2 doesn’t
2 , so we have
belong to PO
(-2,-1)
(-1,1)
(-2,0)
the sequence Γ5 Γ4 Γ3 in the
2
word of the border of PO
(-3,-1)
(-2,1)
(-2,2)
(-4,-1)
(-3,1)
(-2,3)
D’
-1
Period of the naive
−5x−3y
line ⌊
⌋ =0
5
D1
D’
1
(-5,0)
(-4,2)
(-3,4)
(-6,0)
(-5,2)
(-4,4)
(-6,1)
(-5,3)
X
(-6,2)
Γ3
Γ4
Period of the naive
−3x+5y
line ⌊
⌋ =0
5
(-7,1)
(-6,3)
(-7,2)
Γ2
border generated by the sequence LL
Γ1
Γ5
Fig. 1. Border of PO .
(-6,-1)
(-5,1)
(-4,3)
D-1
(-5,-1)
(-4,1)
(-3,3)
D0
(-5,-2)
(-4,0)
(-3,2)
Y
D’
0
(-4,2)
(-3,0)
Γ0
border generated by the sequence LD
Fig. 2. Border of PO2 containing 34 tiles of order 1.
µ1 = s
µ2 = lsq2 −1
if i + 1 is even, µi+1 = µi−1 µqi i +1
if i + 1 is odd, µi+1 = µqi i +1 µi−1
µn is the period.
– As Γ0 is the bottom of mt times this period, the word corresponding to a
short (resp. long) step will be Rq (resp. Rq+1 ). These successive steps give
rise to a vertical segment which corresponds to the letter U . The following
recurrence determines then Γ0 :
ν1 = Rp
ν2 = Rp+1 (U Rp )q2 −1
q
if i + 1 is even, νi+1 = νi−1 νi i+1
qi+1
if i + 1 is odd, νi+1 = νi νi−1
Finally: Γ0 = (νn )(mt) .
c′ x + d′ y
−c′
without the first point of the period which is a neighbour of P1,1 and corresponds to the border Γ4 . Again, the recurrence given by Réveilles [17] to
determine this period induces the recurrence that permits to obtain Γ5 .
– Γ5 is the bottom of mt′ times the period of the digital straight line
Quasi-linear transformations, substitutions, numeration systems and fractals
9
– To obtain Γ3 (resp. Γ2 ) we remark that they are the ”inverse” of Γ0 (resp.
Γ5 ): If we read Γ0 (resp. Γ5 ) from right to left and replace U by D and R
by L (and vice versa) we obtain Γ3 (resp. Γ2 ). Moreover, independently of
a, b, c and d, Γ4 = D and Γ1 = U . Finally Γ0 Γ1 Γ2 Γ3 Γ4 Γ5 corresponds to
the border of PO .
We now consider p-tiles, we have (see proposition 2) POp =
S
p−1 T
X∈PO
(mAbT )X PO :
p−1
p−1
p
we can obtain PO from PO by replacing each point X ∈ PO by PX which is
the image of PO by a given translation (see figure 2). Suppose that the border of
POp−1 contains the letter U , that means that POp−1 contains a point of coordinates
(i, j) such that the point (i + 1, j) doesn’t belong to POp−1 (see figure 2). The
letter U in the border of POp−1 generates the sequence Γ3 (border between Pi,j
and Pi+1,j ) in POp . In the same manner, the letters L, D and R respectively
generate the sequences Γ5 , Γ0 and Γ2 . But this is not sufficient to generate the
border of POp . Indeed, suppose that the border of POn contains the sequence LU ,
this sequence corresponds to the border between three points (i, j), (i − 1, j) and
(i − 1, j + 1) which belong to POp−1 and the point (i, j + 1) which does not belong
to POp−1 (see figure 2), so the sequence LU generates the sequence Γ5 Γ4 Γ3 in the
border of POp . We can see in figure 2 that the sequence LL (resp. LU and LD)
generates the sequence Γ5 Γ4 Γ5 (resp. Γ5 Γ4 Γ3 and Γ5 Γ0 ). So we have to add 12
letters (one letter for each of the 12 sequences of two letters) and to examine
which word has to be associated with these letters. Call L1 , L2 , . . ., L12 these
letters which will be respectively between LL, DL, U L, RR, DR, U R, U U , LU ,
RU , DD, LD and RD. First we have to modify Γi by adding these 12 letters, we
denote Γi′ the new words. In the case studied here (a′ < 0, b′ > 0, | b′ | ≥ | a′ |,
c′ < 0, d′ < 0 and | c′ | ≤ | d′ |) Γ5′ always begins and ends with the letter R
and Γ4′ = D, so the sequence Γ5 Γ4 Γ5 becomes Γ5′ L12 Γ4′ L5 Γ5′ . Now determine the
word generated by L1 : LL1 L generates Γ5′ L12 Γ4 L5 Γ5′ , L generates Γ5′ so L1 will
generate L12 Γ4 L5 . In the same way we examine all other letters and we obtain
a substitution over an alphabet of 16 letters.
10
Marie-Andrée Jacob-Da Col and Pierre Tellier
In figures 3 and 4 we can see two examples of borders of POn . Their corresponding matrix and substitutions are:
Figure
Figure
4 3 0 −1
−1 1
matrix
matrix
1 −1
−1 −1
D →L
D →R
U →R
U →L
L → U L6 RL9 U
L →R
R → DL2 LL11 D R → L
L1 → L6 RL9
L1 → L12 DL5
L2 → L8
L2 → L4
L3 → L4 RL9
L3 → L11 DL5
L4 → L2 LL11
L4 → L8 U L3
L5 → L1 LL11
L5 → L9 U L3
L6 → L12
L6 → L1
L7 → L4 RL4
L7 → L11 DL2
L8 → L6 RL4
L8 → L12 DL2
L9 → L5
L9 → L1
L10 → L1 LL1
L10 → L9 U L6
L11 → L3
L11 → L4
L12 → L2 LL1
L12 → L8 U L6
Fig. 3. Border of PO12 .
4
Quasi-linear transformations and numeration systems
Let β denote a complex number and D a finite set of elements of Z[β]. (β, D)
is a valid base for Z[β] if each element c of Z[β] can be written uniquely in the
Quasi-linear transformations, substitutions, numeration systems and fractals
11
Fig. 4. Border of PO7 .
form
c = c0 + c1 β + c2 β 2 . . . + cn β n
with ci ∈ D and n ∈ N; the length of the decomposition is then n + 1. We also
say that (β, D) is a numeration system of Z[β] and D is the set of digits of this
numeration system.
4.1
Case of Gaussian integers
The results below can be found in [8]: they allow us to determine numeration
systems of Z[i] by considering β = a + ib a Gaussian integer and D a set of
Gaussian integers.
Definition 7 Let c = x+iy
integers.
The integer
and β = a+ib betwo
Gaussian
−bx + ay
c
c
ax + by
+i
.
division of c by β, noted
, is defined by:
=
β
β
a2 + b 2
a2 + b 2
This division corresponds to the usual division of complex numbers composed
with the integer part function: so we have the following relation with QLTs.
Proposition 4. Let
β = a + ib and c = x + iy be two Gaussian integers and let
c
c′ = x′ + iy ′ =
. The point (x′ , y ′ ) is then the image of the point (x, y) by
β
the QLT ⌊gβ ⌋ defined on Z2 by:
2
⌊gβ ⌋ : Z2 −→ Z

ax + by

′

x =
a2 + b 2
(x, y) 7−→
−bx
+ ay


 y′ =
a2 + b 2
12
Marie-Andrée Jacob-Da Col and Pierre Tellier
Theorem 3. Let β = a + ib be a Gaussian integer and D the set of Gaussian
c
= 0, the three following properties are then equivalent:
integers c such that
β
1. (β, D) is a numeration system,
2. The QLT ⌊gβ ⌋ is a consistent Quasi-Linear Transformation,
3. a ≤ 0 and | a | + | b | > 1.
Remark 1. consider the QLT ⌊gβ ⌋ and note PO and POp the tiles defined by
this QLT. The set of digits is given by D = {c = x + iy | ⌊ βc ⌋ = 0} = {c =
x + iy | (x, y) ∈ PO }. In [7] and [9] we can find an algorithm that allows to
determine PO .
4.2
Case of algebraic integers of order 2
Now we consider β an algebraic integer such that β 2 + bβ + a = 0 with a, b ∈ Z.
We only consider the case where β is a complex number, that is to say b2 −4a < 0.
We will define numeration systems of Z[β] where the digits are elements of Z[β].
First we will define an integer division by β, this integer division corresponds
to a QLT that will define the digits of the numeration system. As in preceeding
section, the QLT has to be a consistent QLT. We will define the division by using
another base of Z[β], the QLT associated to the division will depend on these
base, a good choice of the base will raise to a consistent QLT and so defined a
numeration system.
Lemma 1. Let β1 = n + β with n ∈ Z, we have Z[β] = Z[β1 ].
Proof. Let x = x1 + x2 β ∈ Z[β], we have:
⇔ ∃x1 , x2 ∈ Z | x = x1 + x2 β
⇔ ∃x1 , x2 ∈ Z | x = x1 − x2 n + x2 β1
⇔ ∃x′1 , x′2 ∈ Z | x = x′1 + x′2 β1
⇔ x ∈ Z[β1 ]
⊓
⊔
Let now define the integer division using this base. We have β 2 + bβ + a = 0,
−β1 +n−b
so = − β+b
and β12 = β1 (2n − b) − n2 + nb − a. Let x = x1 + x2 β1 ,
a =
a
we have:
1
β
x
(x1 + x2 β1 )(−β1 + n − b)
=
β
a
x2 (β1 (b − 2n) + n2 − nb + a) + β1 (x2 (n − b) − x1 ) + x1 (n − b)
=
a
x1 (n − b) + x2 (n2 − nb + a) + (−x1 − x2 n)β1
=
.
a
We define the integer division of x by β by the composition of the division above
with the integer part function.
Quasi-linear transformations, substitutions, numeration systems and fractals
13
1 n − b a − nb + n2
x1
[
.
−1
−n
x2
a
is defined by x′1 + x′2 β1 ,
Definition 1. Let x = x1 +x2 β1 and Gβ the QLT defined by the matrix
The quotient of the integer division of x by β, noted ⌊ βx
where (x′1 , x′2 ) = Gβ (x1 , x2 ).
The set of digits is
x
D = {x = x1 + x2 β1 | ⌊ ⌋ = 0} = {x = x1 + x2 β1 | G(x1 , x2 ) ∈ PO }
β
where PO is the tile of order one associated to the QLT G.
2
Theorem 4. Let β be an algebraic integer
such that β + bβ + a = 0 and D the
c
= 0, the three following properties are
set of Gaussian integers c such that
β
then equivalent:
1. It exists n such that (β, D) is a numeration system,
2. It exists n such that the QLT ⌊gβ ⌋ (defining the integer division) is a consistent QLT,
3. (b > 2 ) or (b = 2 and
Proof. (β, D) is then a numeration if and only if for all c ∈ Z[β], there exist
c0 , c1 , . . . , cp ∈ D such that
c = c0 + c1 β + . . . + cp β p
Moreover, this decomposition has to be unique.
It is easy
to
see that if this
xi
x
i
decomposition exists then ci = xi + yi β1 with
=G
where x + yβ1 =
yi
y
c. We conclude
exists and is unique if there exists p ∈ N
that
the
decomposition
x
0
p
such that G
=
. Finally (D, β) is a numeration system if and only if
y
0
G is a consistent QLT. Theorem 1 gives the necessary and sufficient conditions
such that G is a consistent QLT but only for QLTs such that ||g||∞ < 1. The
first and the third case of this theorem can not be satisfied. Indeed, in these
two cases we have the condition n2 − nb + a ≥ 0 and we have assumed that
b2 − 4a < 0 so that n2 − nb + a is always positive. The second case corresponds
to the conditions
n − b ≤ 0,
n2 − nb + a > 0,
−1 ≤ 0 and − 1 − n ≤ 0
which is equivalent to −1 ≤ n ≤ b. If we choose n such that ||g||∞ < 1, we have
||G||∞ = a1 max(b − n + n2 − nb + a; 1 + |n|) < 1
⇔
n2 − (b + 1)n + b < 0 (1)
1 + |n| < a (2)
If b ≤ 2, the conditions −1 ≤ n ≤ b and n2 − (b + 1)n + b < 0 are conflicting.
But if b > 2, the condition (1) is satisfied for 1 < n < b and the condition (3)
1 + n < a can always be satisfied (because b2 < 4a).
14
Marie-Andrée Jacob-Da Col and Pierre Tellier
Remark 2. when b = 2 the QLT associated with the integer division is of norm
1. In [7] the conditions for such a QLT to be a consistent QLT are also given,
we can choose n = 1 or n = 2 to obtain a consistent QLT and so to define a
numeration system. In the same way, if n = b or 1 + n = a the QLT is of norm
1; the conditions given in [7] will determine the conditions such that (D, β) is a
numeration system.
Examples: let β 2 + 3β + 3 = 0, and β1 = 1 + β, the
corresponding to
QLT 1 −2 1
the integer division by β is defined by the matrix
and the set of
3 −1 −1
digits is D = {0, −1, −2 − β}, if we choose β1 = 2 + β,the QLT
corresponding
1 −1 1
and the set of
to the integer division by β is defined by the matrix
3 −1 −2
digits is D = {0, −1, −2}.
5
5.1
Quasi-linear transformations and fractals
Border of tiles in 2-dimension
Gilbert [6] proved that if (β, D) is a valid base for Z[β], then every complex
number c ∈ C has an infinite representation (not necessarily unique) in the
form:
p
X
cj β j , cj ∈ D.
c=
j=−∞
Let us consider the set of complex numbers with zero integer part in this
numeration system (also called ”fundamental domain” in the literature [12]),
that is to say the set K of complex numbers z such that:
c=
−1
X
j=−∞
cj β j , cj ∈ D.
In [5] Gilbert determined the fractal dimension of the border of this set con2
p
sidering
the base β
(
) = −n + i with n ∈ N and D = 0, 1, . . . , n . Note K =
p c
P
j
c ∈ C|c =
, the set K is the limit of Kp when p tends toward infinity.
j
β
j=1
√
Denote ρ and θ the module and argument of β: β = ρeiθ . We have ρ = δ
where δ is the determinant of the matrix associated to the division, and so:
(
)
p c
P
j
K p = c ∈ C|c =
with cj ∈ D
j
j=1 β


p


1 X
cj β p−j with cj ∈ D
= c ∈ C|c = p


β j=1


p−1

X
1 
= p c ∈ C|c =
cj β j with cj ∈ D

β 
j=0
Quasi-linear transformations, substitutions, numeration systems and fractals
15
If β is a Gaussian integer D = PO , if β is an algebraic integer
D = {x = x1 + x2 β1 |(x1 , x2 ) ∈ PO } .
In [8] we studied
particular
determinantal QLTs associated with matrices of the
ab
1
and showed how the border of the p-tiles of these QLTs
form a2 +b2
−b a
can be generated by a substitution. This study has been generalized to every
2D determinantal QLT, so we define substitutions that generate the border of
the p-tiles POp . We don’t give the method to generate the substitutions in his
paper, it can be found in []. The susbtitution associated with the QLT (associated with the numeration system), determines the border of √
PO . If at each step
of the substitution we divide the length of these segments δ, we obtain the
border of a fractal set noted F (which corresponds exactely to the fundamental
domain in case of Gaussian integers). Let design by Np the number of segments
of POp , the susbtitution allows to determine Np . Using the counting box dimension we can determine the fractal dimension of the border which is given by
log Np
d = limp→+∞
p . For the examples in section 3 we find a fractal dimension
log δ 2
d = 1, 303 for figure 4 and d = 1, 5236 for figure 3 (which conforms to the result
found in [5]).
5.2
Fractals of Zn
Consider the recurrence given in proposition 2 which allows the construction of
POn ∈ Zn . If we apply this recurrence but starting with a subset PO′ of PO , that
is to say that we define
[
[
′
”
PO′p+1 =
T(mAbT )p X PO′p =
T“mA
d T X PO
′
X∈PO
′p
X∈PO
Property 2. Let denote Np the number of points of PO′p and N the number of
points removed from P0 to obtain PO′ . We have Np = (mn δ n−1 − N )p .
Proof. In property 3 we proved that the number of points of PO equals mn δ n−1
therefore N1 = mn δ n−1 − N . Suppose that Np−1 = (mn δ n−1 − N )p−1 , we
S
have PO′p+1 = X∈P ′ T(mAbT )p X PO′p , it follows that Np = N1 Np = (mn δ n−1 −
O
N )(mn δ n−1 − N )p = (mn δ n−1 − N )p+1 .
⊓
⊔
Like above, at each step of the recurrence, we divide the size of the points
n−1
by mδ n . We then obtain a fractal whose box-counting dimension is given by
d = lim
p→∞
log(mn δ n−1 − N )p
log((mδ
n−1
n
)p )
=
log(mn δ n−1 − N )
log(mδ
n−1
n
)
If we consider numeration systems, PO′ corresponds to a subset D′ of the set of
digits D, so that the fractal obtained corresponds to the set of numbers with
zero integer part and whose decomposition uses only the digits of D′ .
16
Marie-Andrée Jacob-Da Col and Pierre Tellier
Example 1. In figures 5,6 and 7 we see PO and the fractal generated.The black
′
points of P0 are removed to obtain P
O (which
corresponds to the grey points).
0
−3
The QLT in figure 5 is defined by 19
, the fractal dimension of the set
3 0
is log(7)
log(3) = 1, 7712, it corresponds to the numeration system (−3i, {u + iv|u =
0, 1, 2 v = 0, −1, −2}) and represents the set of numbers with zero integer part
and whose decomposition
don’t use the digits 0 and 2− 2i. In figure 6 the QLT is
−2
3
log(8)
1
, the fractal dimension of the set is 2 log(13)
= 1, 6214, it
defined by 13
−3 −2
corresponds to the numeration system (−2 + 3i, {0, −1, −2, −3, −4, −1 + i, −2 +
i, −3 + i − 4 + i, −2 + 2i, −3 + 2I, −2 − i, −3 − i}) and represents the set of
numbers with zero integer part and whose decomposition don’t use the
digits
23
1
0, −4, −1 + i, −3 + 2i and −2 − 2i. In figure 7 the QLT is defined by 9
−2 1
2 log(5)
and the fractal dimension of the set is log(8) = 1, 5479.
Fig. 5. Fractal associated with numeration systems
6
Conclusion
We have seen relations that exists between QLTs, substitutions, numeration
systems and fractals. Quasi-linear transformations generates tilings of Zn and
n-dimensional fractals,we give the fractal dimension of these fractals. We defined
substitutions that generate the border of the tiles in 2 dimension and allow to
compute the fractal dimension of the border. In a future work we will show how
these results can be extended in dimension n. We also used consistent 2D-QLTs
(see definition 1) to define new numeration systems of Z[β] where β is an agebraic
integer of order 2. In [2], the author studied fractals associated with shift radix
Quasi-linear transformations, substitutions, numeration systems and fractals
Fig. 6. Another fractal associated with numeration systems
Fig. 7. Fractal associated with QLT
17
18
Marie-Andrée Jacob-Da Col and Pierre Tellier
systems (which generalize numeration systems). Some of these fractals can be
obtained usign QLTs, one of our future works is to study the relations between
QLTs and these numeration systems.
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