EI, HIROMI

ITO, SHUNJI

Ei, Hiromi

Ito, Shunji

Osaka University Knowledge Archive

Decomposition theorem on invertible substitutions

Osaka City University Repository

Ei, H. and Ito, S.
Osaka J. Math.
35 (1998), 821-834
DECOMPOSITION THEOREM
ON INVERTIBLE SUBSTITUTIONS
HIROMI EI and SHUNJI ITO
(Received April 21, 1997)
0. Introduction
The decomposition theorem of automorphisms of free group is well known, and
we mention the statement in the case of rank 2.
Theorem ([1]). Let G{1,2} be a free group generated by symbols 1 and 2.
Then any automorphism of G{1,2} is decomposed by three automorphisms:
1 -> 2 _ ί l - > 12
/ 3
Recently Zhi-Xiong Wen and Zhi-Ying Wen give the decomposition theorem of
invertible substitutions of rank 2, where we say an automorphism σ is an invertible
substitution if words σ(l) and σ(2) consist of the symbols 1 or 2.
Theorem ([2]). Any invertible substitution is generated by three invertible sub-
stitutions:
In this paper we give a simple proof of the theorem and a geometrical characta-
rization of invertible substitutions.
1. Proof of the theorem
Let us introduce the canonical homomorphism f : G{1,2} —> Z2 as follows:
ί(i±1):=±e
u
 i = l,2
f ( W 0 : = f ( β i ) + f ( « 2 ) +•••+*(«*) for W = Sls2 skeG{l,2}
where {βi,e2} be canonical basis in R2. Then we know the following property.
822 H. Ei AND S. ITO
-1
1
1 1
Fig. 1. IC[W]9 W = 1121- 1 2- 1 1- 1
PROPERTY. Let us define the linear representation L
σ
 of σ by
Then the following commutative relation holds:
G{1,2}4G{1,2}
f I if
A word W e G{1,2} is said to be closed if t(W) = 0. Let V be the family of
polygon curve with integer vertices on iZ2, and let us define the geometrical realiza-
tion map /C : G{1,2} -> V by
and for W =
:= {±\ei I 0 < λ < 1}, i = 1,2
G{1,2}
where x + S = {x + s\s e S}.
If the word W be a closed word, then the definition of IC[W] is modified slightly as
follows:
K[W] := f(17) + K[W
λ
]
where U is the longest word satisfying W = ί7WΛiί7~1.(See Fig. 1.)
Lemma 1. For any automorphism θ, we have
(*) K[Θ{12\-12-1)] = x + q m - ^ " 1 ] for some x G Z 2 .
DECOMPOSITION THEOREM ON INVERTIBLE SUBSTITUTIONS 823
1-1
Fig. 2. 1 )], σ = α, /?, 7
Proof. From Nielsen's theorem, any automorphism σ is decomposed by gener-
ators α, β and 7. On the other hand, it is easy to see that each generator of auto-
morphisms satisfies (*) property. Therefore any composition of generators also has (*)
property. (See Fig. 2.) D
Sublemma 1. Let σ be an invertible substitution and let a linear representation
L
σ
 of σ be
= (a
Assume that άetL
σ
 — ± 1 and max{α, 6, c, d} — 1. Then the invertible substitution σ
is determined by the composition of ay β and 8 as follows:
l - > 1
2 - > 2
1 ->2
2-» 1
1 -> 12
2-> 1
1 -> 12
2 - > 2
(ί
( i
0
(ί
(ί
ί
0
ϊ)
ί)
ϊ)
!)
=> a:
=> β-
= • α ί :
= ^ /9α :
=$aδa :
1
12
\ 2 H 12
or
or
α/3:
Jα :
-{\ 21
2
1
21
2
21
The following sublemma is easily obtained from det L
σ
 = ± 1 .
824 H. Ei AND S. ITO
Sublemma 2. Let ( & d ) be a linear representation of substitution σ. Assume
that detL
σ
 = ±1 and max{α, 6, c, d] > 2 then we have
max{α, 6, c, d} > max{{α, 6, c, d} \ max{α, 6, c, d}}.
Lemma 2. Lef σ be a substitution and let σ(l) and σ(2) be σ(l) = VFi
σ
(2) = W2 Assume that
(1) a //near representation L
σ
 of σ satisfies a> b> d>0 and a > c > d > 0
(2) d e t L
σ
= ± l
(3) K,[σ{\2\-12-1)] =x + QUl-^-1], x E Z2
then there exists non empty word U such that
Wι = W2U or UW2.
Before the proof of the lemma, we give a remark of the assumption (3). The word
σ(121~12~1) is a closed word, therefore K [σ(121"~12~1)] is a closed curve in gen-
eral. And the assumption (3) says that the closed curve consists only of the boundary
of unit square.
Proof. We can introduce the orientation of /C[σ(121~12~1)] naturally by using
the order of symbols in the word. And assume det L
σ
 = 1, then the orientation of
/C[σ(121"12~1)] does not change from the orientation of /C[121"12~1].
(1) The case of W
λ
 = 1W{ and W2 = 2W2.
Suppose iW i^l < 2, where ^V\\ is the length of the word W\, then we can determine
the substitution σ by
1 -> 1 f 1 -> 12
σ :
and these linear representations:
= ( 0 i ° r
This is contradictory to the condition (1).
Let us assume that \W\\ > 3, then W\ and W2 must be decomposed as W\ =
12W[ and W2 = 21W2. By the condition (3) we can easily see from the figure of
/C[σ(121"12-1)] that W
λ
 is decomposed as W
λ
 = UW2 (See Fig. 3.)
(2) The case of W
λ
 = VW{ and W2 = VW^ V φ 0.
Assume that W2 = Q then Wλ is decomposed as W\ — W2U.
Assume that Wi φ 0, then we can find V such that W
λ
 = V1W{' and W2 = V2W2',
DECOMPOSITION THEOREM ON INVHRTIBLE SUBSTITUTIONS 825
(1)
w
(2)
(3)
Fig. 3. K[σ{m-ι2-ιy\
and moreover we see that W" is not empty by the condition (1). Therefore by analo-
gous discussion of case (1) we see that there exist U such that Wi = UW^. (See Fig.
3.)
826 H. Ei AND S. Iτo
We can consider the case of det L
σ
 — —1 by the same manner. D
Lemma 3. Let σ is an invertible substitution which satisfies the condition (I) of
Lemma 2. Then σ can be decomposed by σ = τ o θi (i € {1,2}) with some invertible
substitution r , where 0; is given by
=β
'\2->1 '
Proof. By Lemma 1, the invertible substitution σ satisfies the condition (3) of
Lemma 2 and σ also satisfies the condition (2) from invertibility. So the word W\ is
decomposed as W\ — W2U or UW2 by Lemma 2.
Let us assume that W\ — W2U. Define the substitution r as follows:
(I-+W2
r : \ 2 - > C / '
then we see that σ is decomposed as σ = τoθχ. Both σ and θ\ are invertible, therefore
T is also invertible.
The case of W\ — UW2 is discussed analogously. D
Notice that in the case of Lemma 3 the linear representation LT of r satisfies
c
-
 t j a n c l a - c < a.a b )
a — c\
— a 
Therefore the following relation holds:
max(elements of L
σ
) > max(elements of L
τ
) .
Theorem 1. Any invertible substitution of rank 2 is decomposed by three invert-
ible substitutions:
l-> 12
Proof. Take any invertible substitution σ. By Sublemma 1 if max(elements of
La)— 1 then σ is decomposed by a, β and δ. Consider the case of max(elements of
L
σ
)> 2. By Sublemma 2 we take iχ, jι e {0,1} satisfying
DECOMPOSITION THEOREM ON INVERTIBLE SUBSTITUTIONS 827
α > 6 > d > 0 a n d α > c > d > 0 .
o
By Lemma 3 there exist substitutions τ[ and θPl such that
α*
1
 o σ o or
7
'
1
 = r{ o θPl.
Therefore the substitution σ is decomposed as
σ — a
11
 o r[ o θPl o α j l .
For n := a{l o τ[ let us continue the same procedure. Then there exists τ
n
 such that
max(elements of LTn) = 1, and the substitution σ is decomposed as
σ = τ
n
 o θPn o α
J n
 o o β p 2 o α
 7 2
 o o
where pfc G {1,2} and j f c G {0,1}. D
Let us give a remark related to the uniqueness of decompositions. Define the in-
vertible substitution θ by
θ = /?oαo£ (=δoaoβ).
and replace every substitutions βoaoδ and ί o a o ^ in the decomposition of σ by θ .
Then the substitution σ is decomposed uniquely by a, β, δ and θ in our procedure.
In fact, except the case of W\ = W2UW2 we can determine which we take σ = roθ\
o r σ = r o ί 2 . In the case of W\ = W2UW2, σ can be decomposed as
σ — τoδooίoβ — roβooLoδ.
Using the same discussion, we have the following result.
Theorem 2 (geometrical charactarization of invertible substitutions). Let σ be a
substitution. Then σ is invertible if and only if
/C[σ(121"12-1)] = a; + /C[121-12"1] for some x G Z 2
Proof. If σ is invertible then by Lemma 1
K,[σ(\2\-ι2-1)} = x + K[\2\-ι2-1] for some x G Z 2 .
Oppositely, assume that
(**) /C[σ(121"12-1)] = a: + /C[121-12-1] for x G Z 2
828 H. Ei AND S. Iτo
then we know W\ = W^U or UW2 by Lemma 2. In the case of W\ — W2U (resp.
W\ = UW2) determine the substitution r (resp. r ' ) such that
:\2->C/ i r e S p r * ( 2 -* ί/ J
then σ = τoθ\ (resp. σ = τΌ0 2) and r satisfies (**) property. Continue the procedure,
the substitution σ is decomposed by α, β and 5. So σ is invertible. D
2. Interval exchange transformations and invertible substitutions
In this section, we discuss about the dynamical system called an interval exchange
transformation associated with a substitution.
ASSUMPTION. Let us assume that the substitution σ satisfies the following proper-
ties:
(1) d e t L
σ
 = ± l
(2) the charactaristic polynomial is irreducible.
Let μ be the maximum eigenvalue of L
σ
 and (^) and (1) be column and row
eigenvectors of μ, that is,
Let / be the contracting invariant line of L
σ
, then I is given by
Let li and I2 be unit seguments spanned by ei and e<ι, that is,
h := {λe2 I 0 < λ < 1}
h := {λei I 0 < λ < 1}.
Let us consider a set of unit seguments on lattice points:
S, = - { „ . „ , a-, (h W
We call the union of elements of S^ the stepped curve of the line / and it is denoted
by
S
β
 := (J (x + li).
DECOMPOSITION THEOREM ON INVKRTIBLE SUBSTITUTIONS 829
Let us consider the finite union of Sβ as follows:
«Λ<+oo,(x,l)
λ
GS/3 1
DEFINITION. On the notation of
σ(l) = s1s2 -sk,
σ(2) =
and
let us define a map Σ
σ
 on Q as follows:
for v = 1, 2
Σ
,l
r
) := L~ι(x) + Σ
σ
( β , I
r
) , x E Z
and
The map Σ
σ
 is called the canonical form of σ.
REMARK. The canonical form of σ has another expression, which is for r = 1, 2
Σ
σ
( 0 Λ )
Σ ί-Σ^+^Ol + ί Σ ί-
By the definition of canonical form, Arnoux-Ito ([3]) gives following propositions.
Let U and W be ZY = (ei,Ii) + (e 2 ,I 2 ) and Z/; = (0,li) + (0,! 2). We define the
geometrical realization map K : Q —> {polygons on R2} as follows:
K : (x, l
r
) h^ x + l
r
 for r = 1,2
830 H. Ei AND S. ITO
and let U
a
,β be a projection from R2 to the line I along Q).
Let us define domains, which is finite union of intervals on / in general, as follows:
ΠQ | / 3[K(0,l i)] = D | 0 ) '
Π
β f / ? [K(β i , l < ) ]=Di 0 )
D (o) ._ I I D (o) _ I I D (o) '
u
 *~ Ui=l,2 u i — Ui=l,2 u i
and
Π
β l / 9[K(Σ σ(0,I j))] = D | 1 ) '
Then the following general interval exchange transformation on D^o) and Ό^ are
well-defined:
x i—> x - H
a
ββi if x £ vf'
W{1) : D ^ —-> D^
1)
and the following propositions hold.
Proposition 1 ([3]).
(1) Σ
σ
U D U and Σ
σ
U' D W
Moreover, Σ
σ
U - U = Σ
σ
U' - W.
(2) Assume that (x,h) G Sβ then we have Σ
σ
(a?,Ii) G Q.
(3) Assume that {x,h) Φ {x1 ,\j) t n e n w e n a v e
Proposition 2 ([3]). Let W^i)\
Ό
{o) be the induced transformation ofW^ to the
set Ό(°\ Then we have
(1) ^ ( I ) I D ( O ) = W / ( O )
(2) W^d)|D(o) has σ-structure, that is, for i = 1, 2
C DiJ) for 1 < j < k and WfoΌ™ = D [ 0 ) '
C Dί;> /or 1 < / < i αnrf
= D
DECOMPOSITION THEOREM ON INVERΠBLE SUBSTITUTIONS 831
K[Σ
α
W] K[Σ
β
W] K[Σ
δ
U']
Fig. 4. W{( i )
Using the decomposition theorem in section one, we obtain the following other
charactarization of invertible substitutions.
Theorem 3. A substitution σ is an invertible substitution if and only if the inter-
val exchange transformation Wμ) associated with σ is 2-state interval exchange trans-
formation.
Proof. If σ is an invertible substitution then from the decomposition theorem the
substitution σ is decomposed by the generators a, β and δ. So it is enough to show
that the interval exchange transformations associated with α, β and δ are 2-state inter-
val exchange transformations. (See Fig. 4.)
Oppositely, assume the interval exchange transformation W(\) assosiated with σ is
2-state interval exchange transformation. Without the loss of a generality, we assume
that L
σ
 = ( ft ^ ) satisfies a > b > d and a > c > d by taking α ι o α o α 7 if necessary
832 H. El AND S. ITO
(1) (2) (3) (4)
e
Fig. 5. K[Σ
σ
U']
where i, j G {0,1}.
From the fact that a + b > c + d, that is,
the number of li in K[Σ
σ
W] > the number of 12 in K[ΣσU']
and K[Σ
σ
W] belongs in the stepped curve Sβ from Proposition 1 (2), we see that
there are no (x,l 2) such that (05,12) and (cc-l-ei,l2) G ΣσU\ and ΣσU has the same
property by Proposition 1 (1). Let us consider 4 cases;
• The ends of K[Σ
σ
W] are not constructed by 12 (1)
• One of the ends of K[Σ
σ
W] is constructed by 12 (2) (3)
• Both of the ends of K[Σ
σ
lC] are constructed by 12 (4)
(See Fig. 5.)
The case of (4) is impossible since Σ
σ
U does not contain both (x,h) and (x +
ei, l 2 ) for any x.
For the case of (1) and (2), if (a?,l2) is in ΣσW then (x,li) is also in ΣσW from the
connectedness of K[Σ
σ
U']. So by the definition of Σ
σ
 we have
Then there exists Σ*Lj fs{ s u c n m a t /*, — Y^l=j fSi a n c^ Y^ operating L σ we have
Therefore we have
DECOMPOSITION THEOREM ON INVERTIBLE SUBSTITUTIONS 833
f(tι) = f(sk) and tt = sk.
Continue the same procedure, we obtain
This means that Wi is decomposed as WΊ =
For the case of (3), if (x + e2,b) is in Σ
σ
W then (cc -f ei,l i) is also in Σ
σ
W from
the connectedness of K[Σ
σ
W]. So by the remark we have
Then by the same procedure as the case of (1) and (2), W\ is decomposed as W\ —
W^U. Using same discussion as Lemma 3 in section one, there exists θ{ and r which
decompose σ as σ = r o θi% And notice that
Σ
σ
 = Σ^. o Σ
r
we can say the substitution r also has 2-state interval exchange transformation, since
the interval exchange transformations associated with σ and θi are 2-state interval ex-
change transformations. Continue the same procedure, there exists τ
n
 which satisfies
that
max(elements of LTn) = 1
and we obtain that
σ
 — τ
n
 o θPn o a
jn
 o o ΘP2 o a
J2
 o θPl o a
jl
where pk e {1,2} and j k e {0,1}.
So the substituiton σ is invertible. D
References
[1] Marshall Hall, Jr.: The theory of groups, The Macmillan Company 1959, 90-112.
[2] Z.-X. Wen and Z.-Y. Wen: Local isomorphisms of invertible substitutions, C. R. Acad. Sci.
Paris, 318 Serie I (1994), 299-304.
[3] P. Arnoux and S. Ito: Pisot substitutions and Rauzy fractals, pretirage, Institut de Mathemati-
ques de Luminy, 18 (1998).
834 H. Ei AND S. Γro
H. Ei
Department of mathematics
Tsuda College
2-1-1 Tsudamachi Kodaira
Tokyo, 187-8577, Japan
S. Ito
Department of mathematics
Tsuda College
2-1-1 Tsudamachi Kodaira
Tokyo, 187-8577, Japan


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Decomposition theorem on invertible substitutions

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