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HIGHER DIMENSIONAL EXTENSIONS OF SUBSTITUTIONS AND THEIR

DUAL MAPS

YUKI SANO, PIERRE ARNOUX, AND SHUNJI ITO

TSUDA COLLEGE AND INSTITUT DE MATHEMATIQUES DE LUMINY SEPTEMBER 7, 2000 FINAL VERSION

Abstract. Given a substitution on dletters, we dene itsk-dimensional extension,Ek( ),

for 0 k d. The k-dimensional extension acts on the set of k-dimensional faces of unit

cubes in R

d with integer vertices. The extensions of a substitution satisfy a commutation

relation with the natural boundary operator: the boundary of the image is the image of the boundary. We say that a substitution is unimodular (resp. hyperbolic) if the matrix associated to the substitution by abelianization is unimodular (resp. hyperbolic). In the case where the substitution is unimodular, we also dene dual substitutions which satisfy a similar coboundary condition. We use these constructions to build self-similar sets on the expanding and contracting space for an hyperbolic substitution.

0.

Introduction and statement of results

Let be a substitution on the alphabet W =f1



2

::: d

g. We denote by

A

the linear map on Zdobtained from by abelianization.

For any given point

x

2 Zd, it is natural to associate to each word in W a broken path starting in

x

(associate to the letter

i

the unit segment from

x

to

x

+

e

i, where (

e

1

::: 

e

d) is the canonical basis of Zd, and extend by continuity). We can then de ne a map

E

1( ) on the set of paths, replacing letter

i

by the word (

i

). Care must be taken of the initial point

x

, and it is easily computed that the correct de nition is given by (

x

i

)7!(

A

(

x

)



(

i

)) (see section 2 for the formal de nition). The map

E

1( ) acts in a natural way on the space of formal sums of weighted unit segments.

In this paper, we will de ne higher dimensional extensions

E

k( ) of , acting on formal sums

of weighted

k

-dimensional faces of unit cubes with vertices in Zd. In the case the substitution is unimodular, that is,

A

is an invertible map of Zd, or has determinant +1 or;1, we will also de ne the dual maps

E

k( ), and give explicit formulas. We will prove that these maps commute with the natural boundary morphisms, and establish some basic properties.

Before stating de nitions and results, we wish to give some motivations, since the aim of this work is to set a framework allowing to understand more deeply and generalize previous results.

0.1.

The Rauzy substitution.

In the paper Rauzy], we can nd a curious compact domain

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Let be the Rauzy substitution on three letters de ned by: : 1 !1 2

2 !1 3 3 !1

:

Let

w

= (

w

1

::: w

n

:::

) be the xed point of this substitution,

A

be the linear map associated with by abelianization, P be the contractive invariant plane of

A

, and



:R

3

! P be the projection along the eigenvector corresponding to the maximum eigenvalue



for

A

.

The domain

X

with fractal boundary is obtained as the closure of the set (



Xn k=1

e

wk j

n

= 1



2

:::

)



where f

e

1



e

2



e

3

g is the canonical basis of R

3. This domain is not only interesting in the viewpoint of fractal geometry, but also in the sense of ergodic theory and number theory. In fact, two dynamical systems, a Markov endomorphism with the structure matrix

A

and a quasi-periodic motion, act on the domain

X

, and Rauzy proved that this second dynamical system is measurably conjugate by a continuous map to the dynamical system associated to the substitution , See Rauzy], Ito-Kimura], Ito-Ohtsuki], and Messaoudi].

To study the structure of the domain

X

one of the authors, motivated by dynamical consid-erations, introduced a mapping

E

1( ) on the set

G 1, the Z-module denoted by G 1 = 8 < : X 2 1

n





j

n

2Z

]

f



2 1 j

n

6= 0g

<

+1 9 =  and 1 := Z 3

f1



2



3 g (This set should be interpreted geometrically as the set of formal sums of weighted faces,

i

being the unit face orthogonal to the segment of direction

e

iand (

x

i

)

being the corresponding unit face with lower vertex at

x

+

e

i, so that, for example, (;

e

1



1 ) is the unit face built on

e

2,

e

3 at the origin). The mapping

E

1( ) is de ned by:

E

1( ) : ( ;

e

1



1 ) 7!(;

e

3



1 ) + ( ;

e

3



2 ) + ( ;

e

3



3 )



(;

e

2



2 ) 7!(;

e

1



1 )



(;

e

3



3 ) 7!(;

e

2



2 )



and similar formulae for other (

x

i

).

; ; E1 ( ) 7;! ; ; ; ; ; ; (;

e

1



1 ) ; ; ; ; (;

e

2



2 ) E1 ( ) 7;! ; ; ; ; ; ; (;

e

3



3 ) E1 ( ) 7;! ; ; ; ;

Figure 1. The map

E

1( ) for Rauzy substitution

He proved that, if one considers the element (

x



1 ) as the unit square with sides

e

2,

e

3 with lower vertex in

x

+

e

1, and similarly for the elements (

x



2 ), (

x



3 ), then the sequence of sets

A

n(



(

E

1( ) n((;

e

1



1 ) + ( ;

e

2



2 ) + ( ;

e

3



3 )))) converges to the opposite

;

X

of the set

X

de ned above.

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Moreover, to study the boundary of this set, a mapping

E

2( ) which satis es the following commutative diagram was introduced in Ito-Kimura] and Ito-Ohtsuki] :

G 1 E1 ( ) ;! G 1



# #



G 2 ;! E2 ( ) G 2



whereG

2 is the space of formal sums of weighted edges, and



is the canonical boundary map. We call this mapping

E

2( ) the boundary endomorphism of

E

1( ). The boundary endomorphism

E

2( ) corresponds to the endomorphism on the free group of rank 3 which produces the boundary of fractile discussed by Dekking in Dek1], Dek2]. Using this mapping

E

2( ), the Hausdor dimension of the boundary of

X

is computed in Ito-Kimura].

Recently, Arnoux and Ito showed that for any Pisot substitution on

d

letters, that is, is unimodular and

A

has all eigenvalues, except one, of modulus strictly smaller than one, a compact domain

X

with fractal boundary can be similarly constructed, using the following mapping

E

1( ) on G 1:

E

1( )(

x

i

) = X nj:W (j ) n =i 

A

;1 

x

;

f

(

P

(j) n )

j



where

P

(j)

n is the pre x of length

n

;1 of (

j

), see Arn-Ito].

In this paper, we show how to generalize this construction in any dimension, de ning the map

E

k( ) under suitable hypotheses.

0.2.

Geometric models for substitutions.

We can associate to any substitution a map

E

1( ) on 1 dimensional broken paths in R

3, as explained above it is immediately checked that the image by this map of a closed path is a closed path. One can then ask whether one could de ne a map on faces, which sends a face with boundary



to a union of faces with boundary

(



) (See gure 2.3 for the example of Rauzy substitution).

In this paper, we introduce the higher dimensional mapping

E

2( ) called 2-dimensional ex-tensionof which solves this problem. It is de ned on the set

G 2 := 8 < : X 2 2

n





j

n

2Z

]

f



2 2 j

n

6= 0g

<

+1 9 =  where 2 :=

Zdf

i

^

j

j 1

i < j

d

g (one should think ofG

2 as the set of formal sums of weighted 2-dimensional faces). In fact, we will solve the problem in all dimensions, see theorem 2.1.

We will show that the mapping

E

1( ) (resp.

E

2( )) de ned above is the dual, in the ordinary sense, of the linear mapping

E

1( ) (resp.

E

2( )), showing that the de nition, which seems com-plicated, is in fact natural these dual maps also satisfy commutation relation with a suitably de ned coboundary operator, see theorem 3.1.

0.3.

The theorem of EI.

In the paper Ei-Ito], it is proved that a substitution on two letters 1



2 is invertible, that is, it extends to an automorphisms of the free group on two elements, if

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and only the words (12) and (21), which have the same length, dier only in two consecutive indices, where one word contains 12 and the other 21. Graphically, this means that the paths associated as above to (12) and (21) dier only on the boundary of a unit square.

We can rephrase it as saying that the substitution is invertible if and only if the mapping

E

2( ) de ned below sends a square exactly to a square (with opposite orientation if the deter-minant of the substitution is -1) it is tempting to ask whether this result can be generalized to more letters, and we will provide a partial answer below.

0.4.

Structure of the paper.

In section 1, we x notations, de ne the spaces Gk of formal sums of weighted

k

-dimensional faces and their dual spaces Gk, and de ne the boundary and coboundary operators. We also de ne an isomorphism

k between Gk and Gn

;k, similar to Poincare duality, which allows in suitable cases to give a geometric meaning to elements ofGk. In section 2, we de ne the main object of this paper, the

k

-dimensional extension

E

k( ) of

a substitution ,and we show that the

k

-dimensional extensions satisfy a commutation relation with the boundary operator (theorem 2.1) this theorem solves the problem given in section 0.2. In section 3, we show that, under suitable hypothesis (namely, the matrix

A

has determinant one), we can compute explicitly the dual map

E

k( ) of

E

k( ), and that these maps also satisfy

a commutation relation with the coboundary operator.

In section 4, we show that, in the case of hyperbolic substitutions (when

A

has no eigenvalue of modulus 1), we can use this construction to de ne, by iteration and renormalization, a self similar set in the stable space and in the unstable space in the particular case of so-called Pisot substitutions, this can be used to build a geometric model for the dynamical system associated to the substitution, and a Markov partition for the toral automorphism

A

. This allows us to generalize the construction of the Rauzy fractal. In section 5, we give a weak generalisation of the theorem of Ei: a substitution on

d

letters gives rise to an automorphism of the free group only if its top-dimensional extension associates a cube to a cube.

In the nal two sections, we give a few explicit examples, and raise some open questions and possible generalizations.

A nal remark: the de nition of the upper-dimensional extensions is more or less natural from the statement of the problem it solves the reason why we consider the dual maps is much less clear at rst sight. We can give two main motivations:

First of all, the dynamical system associated to a substitution on the alphabet W has very speci c property: it is a self-induced system more speci cally, it is a system

T

:

X

!

X

, together with a generating partition f

X

ij

i

2 Wg, and a subset

A

with partition f

A

ij

i

2 Wg, such that the induced map

T

A of

T

on

A

is conjugate to

T

, and, for any point

x

2

A

, the symbolic dynamics of

x

under

T

with respect to the

X

iis obtained from the symbolic dynamics

of

x

under

T

A with respect to the

A

i by applying the substitution . Rauzy had rst the idea

to generate such a system by an\exduction" process, that is, starting with an arbitrary system, and building a larger systems from which it comes by induction, the symbolic dynamics with respect to suitable partitions being related by the given substitution. If one can, by a suitable renormalization, obtain convergence to a self-induced dynamical system, one obtains a geometric model for the substitution. Rauzy was able to build this for speci c examples, and later the third author of this paper gave a general formula to build the \exduction" it turns out that this formula is exactly the dual map of the one-dimensional extension.

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Second, it is easy to prove that, for an hyperbolic substitution with unstable space of di-mension

k

, the iterates of a

k

-dimensional face by

E

k( ) stay within bounded distance of the

unstable space. We can thus obtain a discrete approximation of the stable space. To approx-imate the stable space, it would be natural to study the inverse map of

E

k( ). However, the

k

-dimensional extension is not invertible in general but the dual map is always de ned, and can replace the inverse map. One can check below that, in the case of unimodular substitutions, the map

E

0( ) is unitary with repect to the natural quadratic form on

G

0, so that in that case the dual and inverse maps are isomorphic (this is also the case for

A

) this does not hold in the higher dimensional case, but then the non-existant inverse map can be replaced by the dual map.

Acknowledgement. We thank the anonymous referee for his many useful comments.

1.

Framework and notations

1.1.

The substitution.

We consider an alphabet W =f1



2

::: d

g. The free monoid on W, or the set of nite words on W is denoted byW =

S 1

n=0

f1



2

::: d

gn.

A substitution on W is a non-erasing morphism of the free monoid. It is completely de ned by its value on the letters, that is, by a map from W to W which takes each letter to a non empty word, and extends in a natural way to an endomorphism on W by the rule

(

U

) (

V

) = (

UV

), for

UV

2W . It also extends to a map on the setW

Nof in nite sequences with value in W, and to an homomorphism on the free group on W.

Notations 1.1.

In the sequel, we take the following notations (

i

) =

W

(i)=

W

(i) 1

W

(i) li where

l

i is called the length of (

i

). We also write

(

i

) =

W

(i)=

P

(i) n

W

(i) n

S

(i) n where

P

(i) n =

W

(i) 1

W

(i)

n;1 is the prex of length

n

;1 of the word

W

(i)(empty word for

n

= 1), and

S

(i)

n =

W

(i)

n+1

W

(i)

li is the sux of length

l

i

;

n

of

W

(i) (empty word for

n

=

l

i).

Letf

e

1

::: 

e

d

gbe the canonical basis ofZd. We will denote by

f

the natural homomorphism (abelianization) from W to Zd given by

f

(

i

) =

e

i for all symbols

i

2W. For any nite word

W

2W ,

f

(

W

) =t(

x

1

::: x

d), where

x

i is the number of occurrences of the letter

i

in

W

. There exists a unique linear transformation

A

satisfying the following commutative diagram:

W ;! W

f

# #

f

Zd ;!

A Zd

:

Remark that the matrix of

A

, as a linear map on Zd, is by construction a positive matrix. This is the dierence between substitutions and the general case of endomorphisms of free groups most of what we do could in fact be extended to such an endomorphism.

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1.2.

Faces of dimension k and boundary morphisms.

We de ne the symbolic sets

A

k by

A

0 := f g and

A

k :=f

i

1 ^

i

2 ^^

i

k j1

i

1

< i

2

< i

k

d

g for 1

k

d:

Let k (0

k

d

) be the following product spaces:

k :=Zd

A

k

:

One should think of an element (

x

i

1

^^

i

k) of k as the face of dimension

k

, along unit vectors

e

i1

::: 

e

ik, of the unit cube at the origin in

Rdtranslated by

x

2Zd, and of an element (

x



) of 

0 as the point

x

2Zd.

Denition 1.1.

We denote by Gk the freeZ-module with as generators the elements of k: Gk := 8 < : X 2 k

n





j

n

2Z

]

f



2kj

n

6= 0g

<

+1 9 = 

:

We think ofGk as the space of formal nite sums of weighted faces, with weight in Z. The element (

x

i

1

^^

i

k) has been de ned only in the case where

i

1

< i

2

<

< i

k. We will take advantage of the notation to de ne it in the general case, by (

x

i

1

^^

i

k) = 0 if

i

n=

i

mfor some

n

6=

m

, and (

x

i

 (1)

^^

i



(k)) =

(

)(

x

i

1

^^

i

k) otherwise, where

(

) is the signature of the permutation

off1

::: k

g, so that we recover antisymmetry of the wedge product for example, (

x

i

1

^

i

2) = ;(

x

i

2 ^

i

1).

We will say that an element of Gk is geometric if all its non-zero coecients are +1 or ;1. The simplest geometric elements are the elements of k. To an element (

x

i

1

^^

i

k), we can associate its geometric realization, that is, the setf

x

+

t

1

e

i 1+

+

t

k

e

i k

j0

t

n 1



1

n

k

g together with the orientation given by the basis

e

i1

:::

e

ik. For an arbitrary geometric element P

2

k

n





, we de ne its geometric realization as the union of the geometric realizations of the elements



such that

n

6= 0. In particular, the geometric realization of 0 is the empty set.

Since it should not lead to confusion, we will denote a geometric element and its realization in the same way.

; ;  (

x



1)  ; ; ;(

x



1)

e

1 ; ; -; ;   (

x



1^2) ; ; ; ; ; ; ; ; (

x



1^2^3) -; ;   ; ;  ;(

x



1^2)

Figure 2. Examples of faces with 3 letters We can now de ne boundary maps:

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Denition 1.2.

The boundary maps



k : Gk ! Gk ;1 (1

k

d

) are dened on basis vectors by:



k(

x

i

1 ^^

i

k) := k X n=1 (;1)nf(

x

i

1 ^^ b

i

n^^

i

k) ;(

x

+

e

i n

i

1 ^^ b

i

n^^

i

k)g



where as usual,

i

1 ^^ b

i

n^^

i

k =

i

1 ^^

i

n ;1 ^

i

n +1

^^

i

k and in the case

k

= 1, we put ^

i

= .

In particular,



1 is given by:



1(

x

i

) =

;(

x



)+ (

x

+

e

i



)

:

We extend



k to all ofGk by linearity it is straightforward to check that the boundary maps satisfy the usual relations



k;1



k = 0.

1.3.

Duality.

One can de ne in the obvious manner a dual space since we are in an in nite-dimensional Z-module, this de nes a complicated space, and we will restrict ourself to the set of dual maps with nite support. We will denote this set by Gk it has a natural basis, and we will denote by (

x

i

1

^^

i

k) the dual vector of (

x

i

1

^^

i

k). De ning the sets

A

k by

A

k :=f

i

1 ^

i

2 ^^

i

k j1

i

1

< i

2

< i

k

d

gfor 1

k

d

and k :=Zd

A

k, we see thatGk is the freeZ-module generated by the elements of k.

We will writehithe natural product betweenGk and its dual, that is, for an element

F

of Gk and

of the dual, we de ne h

F

i=

(

F

).

We can get the formula of dual boundary maps



k from Gk toGk +1:

Proposition 1.1.

The dual boundary maps



k :Gk

;1

!Gk (1

k

d

) are given as follows:



k(

x

i

1 ^^

i

k ;1) = d;k+1 X n=1 (;1) jn;n+1 f(

x

i

1 ^^

i

j n ;n ^

j

n^

i

j n ;n+1 ^^

i

k ;1) ;(

x

;

e

j n

i

1 ^^

i

j n;n ^

j

n^

i

j n;n+1 ^^

i

k ;1) g



where f

j

1

j

2

::: j

d;k+1 g=Wnf

i

1

i

2

::: i

k;1 g, with

i

1

<

< i

k ;1,

j

1

<

< j

d ;k+1, and

i

jn;n

< j

n

< i

jn;n+1. In the case

k

= 1, we put

f g:= , hence



1 is given by



1(

x



) = d X i=1 f(

x

;

e

i

i

);(

x

i

)g

:

Proof. This is simple computation using the fact that, by de nition, we have

< 

k(

x

i

1 ^^

i

k ;1)



(

y

j

1 ^^

j

k)

>

=

<

(

x

i

1 ^^

i

k ;1)



k(

y

j

1 ^^

j

k)

> :

The only non-trivial part is the fact that

i

jn;n

< j

n

< i

jn;n+1. Recall that the nite se-quences (

i

1

::: i

k;1) and (

j

1

::: j

d;k+1) are increasing, and complementary of each other in f1



2

::: d

g. Consider the setf1



2

::: j

n;1g by de nition, it contains

j

n;1 elements, and among these,

n

;1 elements belong to (

j

1

::: j

n;1). Hence, the remaining

j

n

;

n

elements belong to (

i

1

::: i

k;1), which proves that

i

j

n

;n

< j

n

< i

j n

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It is easily computed that

A

k and

A

d;k have the same cardinal, and we can use this to de ne linear isomorphisms between Gk and Gd

;k:

Denition 1.3.

For 0

k

d

, we dene maps

'

k from Gk to Gd ;k by:

'

k(

x

i

1 ^^

i

k) := (;1)i 1+i k(

x

+

e

i1+ +

e

i k

j

1 ^^

j

d ;k)



where f

i

1

:::i

k g and f

j

1

:::j

d;k

g form a partition of f1



2

::: d

g, with

i

1

<

< i

k,

j

1

<

< j

d

;k. In the case

k

= 0, we put ( ;1) := 1,

e

 :=

0

, and f g:= , hence

'

0 is given by

'

0(

x



) := (

x



1^2^^

d

)

:

The map

'

k is one-to-one. It is possible to de ne as above geometric elements in the dual,

and one can check that

'

k sends a geometric element of the dual to a geometric element. We can

thus de ne the geometric realization of a geometric dual element as above, we will not make a dierence between an element and its geometric realization.

A straightforward computation, left to the reader, shows that the duality isomorphisms con-jugate boundary and coboundary:

Proposition 1.2.

The dual boundary maps



k :Gk ;1

!Gk (1

k

d

) satisfy the equality:

'

k



k =



d ;k+1

'

k ;1

:

2.

Denition of higher dimensional extensions and commutation with the

boundary operators

2.1.

Denition of

E

0( )

and

E

1( )

.

The unique linear map

A

on

Zd that commutes with the abelianization extends in a natural way to a map

E

0( ) on

G

0 by

E

0( )(

x



) := (

A

(

x

)



). We want to de ne a map onG

1. It suces to de ne it on basis elements (

x

i

), and to extend by linearity. Recall the informal motivation given at the beginning of the paper: to the basis element (

x

i

) we associate a unit segment along vector

e

i starting at

x

. We want to de ne

E

1( ) in such a way that it sends (

x

i

), to some path associated to the word (

i

). For the image of a continuous path associated to

ij

to be a continuous path, we need also to act on the origin of the segment, which leads, on the notation (

i

) =

P

(i)

n

W

(i)

n

S

(i)

n , to the following de nition:

Denition 2.1.

The map

E

1( ) is the unique linear map on G 1 dened by:

E

1( )(

x

i

) = li X n=1 

A

(

x

) +

f

(

P

(i) n )

W

(i) n

:

It is easy to check that the image of a geometric element corresponding to a path from

x

to

y

is a path from

A

(

x

) to

A

(

y

). This is a geometric way to visualize the abelianization map. In particular, taking negative signs to take into account the orientation, we can consider closed paths, whose images will again be closed paths.

An other remark is that we can extend

E

1( ) to elements with in nite support, in particular to in nite paths. If we consider an in nite sequence which is a xed point of the substitution, the corresponding path from the origin is xed by

E

1( ). This can be a interesting way to represent this xed sequence.

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2.2.

Denition of

E

k( )

.

We want now to de ne linear maps

E

k( ) on Gk, for all

k

. By linearity, it is enough to de ne them on basis elements, and the following de nition encompasses the previous one for

E

1( ).

Denition 2.2.

We dene the higher dimensional extension of dimension

k

of , denoted by

E

k( ), for 1

k

d

, by:

E

k( )(

x

i

1 ^^

i

k) := li 1 X n1 =1 li k X nk =1 

A

(

x

) +

f

(

P

(i 1 ) n1 ) + +

f

(

P

(i k ) nk )

W

(i 1 ) n1 ^^

W

(i k ) nk

where we use the antisymmetry of the wedge product: (

y

j

1

^^

j

k) = 0 if there exists

l

6=

m

such that

j

l =

j

m and otherwise, (

y

j

(1)

^^

j



(k)) =

(

)(

y

j

1

^^

j

k).

2.3.

Commutation with the boundary operator.

We will show in this section that the maps de ned above solve one of the questions of the introduction. Namely, the image of the boundary of a

k

-dimensional face is the boundary of the image of this face, that is, the maps

E

k( ) commute with the boundary operators.

Theorem 2.1.

The following commutative diagram holds: Gd Ed ( ) ;! Gd



d # #



d Gd ;1 Ed;1 ( ) ;! Gd ;1



d;1 # #



d ;1 ... ... G 1 E1 ( ) ;! G 1



1 # #



1 G 0 E0( ) ;! G 0

:

Proof. We need to show the relation



k

E

k( )(

x

i

1 ^

i

2 ^^

i

k) =

E

k ;1( )



k(

x

i

1 ^

i

2 ^^

i

k) (2.1) For

k

= 1, it is easily checked in that case, it just means that the image of a path from

x

to

y

is a path from

A

(

x

) to

A

(

y

), which was the initial purpose of the de nition of

E

1( ). We will prove the relation for

k

= 2, leaving the general case to the reader.

The right hand side of the relation is given by

E

1( )



2(

x

i

1 ^

i

2) =

E

1( )(

x

i

1) ;

E

1( )(

x

+

e

i 2

i

1) ;

E

1( )(

x

i

2) +

E

1( )(

x

+

e

i 1

i

2) = li 1 X k=1 

A

(

x

) +

f

(

P

(i1) k )

W

(i1) k ; 

A

(

x

+

e

i2) +

f

(

P

(i1) k )

W

(i1) k ; li 2 X k=1 

A

(

x

) +

f

(

P

(i 2 ) k )

W

(i 2 ) k ; 

A

(

x

+

e

i1) +

f

(

P

(i 2 ) k )

W

(i 2 ) k

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The left hand side of the relation is given by



2

E

2( )(

x

i

1 ^

i

2) =



2 2 4 li 1 X j=1 li 2 X k=1 

A

(

x

) +

f

(

P

(i1) j ) +

f

(

P

(i2) k )

W

(i1) j ^

W

(i 2 ) k 3 5 = li 1 X j=1 li 2 X k=1 h

A

(

x

) +

f

(

P

(i 1 ) j ) +

f

(

P

(i 2 ) k )

W

(i 1 ) j ; 

A

(

x

) +

f

(

P

(i 1 ) j ) +

f

(

P

(i 2 ) k ) +

f

(

W

(i 2 ) k )

W

(i 1 ) j i ; li 1 X j=1 li 2 X k=1 h

A

(

x

) +

f

(

P

(i 1 ) j ) +

f

(

P

(i 2 ) k )

W

(i 2 ) k ; 

A

(

x

) +

f

(

P

(i 1 ) j ) +

f

(

P

(i 2 ) k ) +

f

(

W

(i 1 ) j )

W

(i 2 ) k i

Since we have, for 1

k < l

i 2,

f

(

P

(i 2 ) k ) +

f

(

W

(i 2 ) k ) =

f

(

P

(i 2 )

k+1) by de nition, we see that, in the rst sum, for xed

j

, all terms except the rst and last cancel in pairs since we have

f

(

P

(i2) li 2 ) +

f

(

W

(i2) li 2 ) =

f

( (

i

2)) =

A

(

e

i

2), the rst sum reduces to

li 1 X j=1 

A

(

x

) +

f

(

P

(i1) j )

W

(i1) j ; 

A

(

x

+

e

i2) +

f

(

P

(i 1 ) j )

W

(i1) j

:

A similar argument for the second sum, xing

k

and varying

j

, proves both sides of the relation 2.1 are equal.

The rst non-trivial example is given by the Rauzy substitution, see gure 2.3.

3.

Dual maps for unimodular substitutions, and coboundary

3.1.

The dual map.

Since

E

k( ) are linear maps, we can de ne their dual maps on the dual

spaces ofGk. For an element

F

ofGk and

of the dual, the dual map is de ned by the relation h

FE

k( )

i=h

E

k( )

F

i.

Note that, if

A

is not one-to-one, there can be in nitely many elements of k whose image

contain a given element, since the equation

A

(

y

) +

f

(

P

(j)

n ) =

x

can have in nitely many

solutions. Hence, in general, the space of dual elements with nite support is not invariant by

E

k( ).

If however the map

A

is one-to-one, this argument proves that the spaceGk of elements with nite support is invariant by

E

k( ), and it makes sense to compute explicitly this map for the canonical basis.

3.2.

Explicit computation of the dual map: the unimodularity condition.

A substitu-tion is said to be unimodular if its abelianizasubstitu-tion

A

has determinant +1 or;1.

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; ;  E1 ( ) 7;! ; ;  -(

0



1) -; ; (

0



2) E1 ( ) 7;! ; ;  6 ; ; 6 (

0



3) E1 ( ) 7;! ; ;  ; ; ; ; E2( ) 7;! ; ; ; ; ; ; ; ; (

0



1^2) ; ; ; ; (

0



1^3) E2( ) 7;! ; ; ; ; ; ; ; ; (

0



2^3) E2 ( ) 7;! ; ; ; ; ; ; ; ; ; ; E2( ) 7;! ; ; ; ; ; ; (

0



1^2) ?



2 ?



2 ; ;  -; ;   E1 ( ) 7;! ; ;  ? -; ;  6

Figure 3. The gure of

E

k( ),

k

= 1



2 and the commutation with boundary for Rauzy substitution

To compute the image of a basis element of the dual, we must calculate its value on a basis element ofGk, which is given as follows:

<

(

y

j

1 ^^

j

k)

E

k( )(

x

i

1 ^^

i

k)

>

=

< E

k( )(

y

j

1 ^^

j

k)



(

x

i

1 ^^

i

k)

>

=Xk p=1 lj p X np=1

<

(

A

(

y

) +Xk p=1

f

(

P

(jp) np )

W

(j1) n1 ^^

W

(j k ) nk )



(

x

i

1 ^^

i

k)

> :

The product takes value 0 except if one of the faces in the left-hand side corresponds to the dual element on the right hand side, that is, if we can nd indices

n

p

p

= 1

::: k

such that:

A

(

y

) +Xk p=1

f

(

P

(j p ) np ) =

x

and the face

W

(j

1 ) n1 ^^

W

(j k )

nk is equal, up to orientation, to the face

i

1

^^

i

k that is, there exists a permutation

of f1

::: k

g such that,for all 1

l

k

W

(j l

)

nl =

i

 (l)

:

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In that case, we have

<

(

A

(

y

) +Xk p=1

f

(

P

(j p ) np )

W

(j 1 ) n1 ^^

W

(j k ) nk )



(

x

i

1 ^^

i

k)

>

=

(

)

In general, this can lead to a complicated case study, since

A

needs not be invertible onZd. If however the substitution is unimodular,

A

can be inverted, and the computation above gives an explicit formula for the dual.

Proposition 3.1.

If the substitution is unimodular, the mappings

E

k( ) (1

k

d

) are given on Gk by:

E

k( )(

x

i

1 ^^

i

k) = X 2S k X 1lk X W(j l ) n l =i (l)

(

)

A

;1 

x

;

f

(

P

(j 1 ) n1 ) ;;

f

(

P

(j k ) nk )

j

1 ^^

j

k

:

In the special case

k

= 0, the formula also makes sense, and it is worth to remark that

E

0( ) is also the inverse of

E

0( ), as was already noted at the end of the introduction. For

k

= 1, we obtain the simpler formula already given in the introduction:

Corollary 3.1.

If the substitution is unimodular, the mapping

E

1( ) is given by:

E

1( )(

x

i

) = X nj:W (j ) n =i 

A

;1 

x

;

f

(

P

(j) n )

j

:

3.3.

Commutation with coboundary.

We get the following relation among dual mappings, using the commutation relations proved in the previous section, and the property of a dual mappping:

Theorem 3.1.

The following commutative diagram holds: G 0 E0 ( ) ;! G 0



1 # #



1 G 1 E1 ( ) ;! G 1



2 # #



2 ... ...



d;1 # #



d ;1 Gd ;1 Ed;1 ( ) ;! Gd ;1



d # #



d Gd Ed ( ) ;! Gd

:

Proof. From Theorem 3.1,



k

E

k( ) =

E

k

;1( )



k. Using the property of composition of dual mappings, (



) =



, we have

E

k( )



k =



k

E

k

;1( ).

3.4.

Geometric interpretation of dual mappings.

Recall that we can identify dual spaces Gk to Gd

;k, using isomorphisms

'

k :

Gk ! Gd

;k given by de nition 1.3. If we conjugate dual mappings

E

k( ) on Gk by these isomorphisms, we obtain mappings on Gd

;k. A straightfor-ward computation, using proposition 1.2, proves that these mappings commute with the usual boundary morphism.

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; ; E1 ( ) 7;! ; ; ; ; ; ; (;

e

1



1 ) ; ; ; ; (;

e

2



2 ) E1 ( ) 7;! ; ; ; ; ; ; (;

e

3



3 ) E1 ( ) 7;! ; ; ; ; 6 ; ; E2 ( ) 7;! -? ; ; ;(;

e

1 ;

e

2



1 ^2 ) ; ;  ;(;

e

2 ;

e

3



2 ^3 ) E2 ( ) 7;! ; ; 6 ; ; -(;

e

1 ;

e

3



1 ^3 ) E2 ( ) 7;! ; ;  ? E1 ( ) 7;! ; ; ; ; ; ; (;

e

1



1 ) ?



2 ?



2 6 -?  E2 ( ) 7;! -? ; ;   6 ; ; 

Figure4. The gures of

E

k( ),

k

= 1



2 and the commutation with coboundary for Rauzy substitution

An application of this property is the following: in some cases, we are interested in the geometric set associated to

E

1( )

n(

x

i

) in particular, we want to study its boundary, and this

is just the geometric set associated to

E

2( )

n(

x

i

) this can give an easy way to study this

boundary, and in particular its dimension.

The gures 2.3 and 3.4 show, in the case of Rauzy substitution, the commutative relations with the boundary map



2 and the coboundary map



2 this answers the questions of sections 0.1 and 0.2.

4.

Hyperbolic substitutions: Hausdor convergence of renormalized iteration

Denition 4.1.

A substitution is said to be hyperbolic if it is unimodular and its abelianization

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In that case, the space Rn splits into a pair of invariant spaces: the stable space

E

s, where the restriction of the map

A

is strictly contracting for an appropriate norm, and the unstable space

E

u, where

A

is strictly expanding we will denote by



s (resp.



u) the projection on

E

s

along

E

u (resp. on

E

u along

E

s).

Let us consider an hyperbolic substitution, with an unstable space of dimension

k

. Remark that, since

A

is strictly expanding on

E

u, it is one-to-one from

E

u to itself. Hence, without

any hypothesis,

A

;1, as a linear map on the real vector space

E

u, is well de ned (even if

A

is

not invertible as a map onRd).

Theorem 4.1.

Let be an hyperbolic substitution, with an unstable space of dimension

k

. For any

k

-dimensional face (

x

i

1

^^

i

k), the sequence of compact subsets of

E

u:

X

n=

A

;n

(



u(

E

k( )n(

x

i

1

^^

i

k))) converges in the sense of Hausdor .

Proof. The main ingredient in the proof is the following lemma, whose proof is left to the reader:

Lemma 4.1.

Let

ABCD

be 4 compact sets in Rd. The Hausdor metric satises

d

h(

A



BC



D

) max(

d

h(

AC

)

d

h(

BD

)).

A simple computation shows that the Hausdor distance between



u(

x

i

1

^

:::

^

i

k) and

A

;1(



u(

E

k( )(

x

i

1

^^

i

k))) is bounded by a constant

K

independent of

x

, since the renor-malization by

A

;1 cancels the

A

that occurs in the de nition of

E

k( ).

But we know, by hypothesis, that

A

;1 is strictly contracting with ratio

 <

1 for a suitable norm. Using the lemma, we immediately obtain

d

H(

X

n

X

n+1)

K

n. The distance decreases exponentially fast, hence the sequence (

X

n)n2Nis a Cauchy sequence in the Hausdor topology. But it is well known that the space of compact subsets of Rd is complete for the Hausdor topology. Hence the sequence (

X

n)n2Nconverges.

A similar theorem can be proved for the dual maps in the unimodular case:

Theorem 4.2.

Let be an unimodular hyperbolic substitution, with a stable space of dimension

d

;

k

. For any

k

-dimensional dual face (

x

i

1

^^

i

k), the sequence of compact subsets of

E

s

X

n=

A

n(



s(

E

k( )n(

x

i

1

^^

i

k))) converges in the sense of Hausdor .

The proof is exactly similar to the preceding one, the only dierence being that it is now

A

, and not

A

;1, that is contracting.

We can get deeper results in a special case: we say that a substitution is Pisot if it is uni-modular and all its eigenvalues, except one, are of modulus strictly smaller than one.

In that case, the stable space is of codimension 1 the results above prove that the sets

X

(i) = lim

n!1

A

n(



s(

E

1( )

n(

0

i

)) are well de ned. However, although, for xed

n

, the sets



s(

E

1( )

n(

0

i

) are disjoint, up to sets of measure 0, it is unclear that this property holds in

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Denition 4.2.

We say that the substitution has immediate coincidence for all letters if, for all pairs of letters

ij

, there is an index

k

such that:

W

(i)

k =

W

(j)

k and

f

(

P

(i)

k ) =

f

(

P

(j)

k )

The geometric meaning of this condition is that, if one represents each word (

i

) as a broken path starting from 0, every two paths share at least one edge.

In the paper Arn-Ito], it is proved that, under this hypothesis, the domains

X

(i)are pairwise disjoint sets up to sets of Lebesgue measure 0, and the following theorem is proved:

Theorem 4.3.

Let be a unimodular Pisot substitution satisfying the coincidence condition. The dynamical system associated to the substitution is measurably conjugate by a continuous map to a domain exchange dened on the sets

X

(i).

5.

Invertible substitution and top-dimensional extensions

In the maximal dimension,

k

=

d

, there is only one type of face: the unit cube of dimension

d

. The corresponding map

E

d( ) associates to each unit cube a nite sum of weighted unit cubes,

and, abbreviating

C

= 1^2^^

d

, can be expressed as

E

d( )(

x

C

) = P

y 2Z d

n

y(

y

C

). A straightforward computation shows the following:

Proposition 5.1.

For any substitution, the sum P y 2Z

d

n

y is equal to the determinant of the map

A

.

In particular, for a unimodular substitution, this sum is +1 or;1.

Denition 5.1.

A substitution is said to be invertible if it extends to an automorphism of the free group.

Invertible substitution on 2 letters are completely characterized, see Wen-Wen] and Mig-See] in particular, they are sturmian, that is, they preserve sturmian words, or words of minimal complexity.

It has been shown in Ei-Ito] that a necessary and sucient condition for , on two letters, to be invertible, is that the words (

ij

) and (

ji

) dier only in two consecutive places. If we consider the boundary of the unit square at the origin, and take into account the boundary relations, this exactly means that the substitution is invertible if and only if the image by

E

2( ) of a unit square is a unit square (with weight +1 or ;1). This result can be partially generalized in any dimension:

Proposition 5.2.

Let be an invertible substitution on

d

letters if we denote

E

d( )(

x

C

) by

P y 2Z d

n

y(

y

C

), we have P y 2Z d j

n

y j= 1.

Proof. We just remark that, from a straightforward computation,

E

d(

) =

E

d( )

E

d(

). The

generators of the group of automorphisms of the free group are known (see MaKaSo]), and it is an easy exercice to check that the property is true for all the generators, hence for all the automorphisms of the free group.

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Another way to say it is that, if is invertible, the image of a unit cube by

E

d( ) is exactly

one cube (with positive or negative orientation).

For the end of this proof, we need to check that the de nitions we gave for the substitutions are still valid for homomorphisms of free groups, see the last section. A direct proof restricted to substitutions would be much more dicult, since the structure of the monoid of invertible substitutions seems to be quite complicated, except for two letters where it is completely known.

It would be interesting to know if this necessary condition is also sucient.

6.

Examples

Example

1

.

Let be the substitution

: 1 !121 2 !12

:

This is called Fibonacci substitution. The matrix

A

and the inverse matrix

A

;1 are

A

= 2 1 1 1 

and

A

;1 = 1 ;1 ;1 2 

:

Then the linear mapping

E

1( ) and the dual mapping

E

1( ) are given by

E

1( ) : (

0



1) 7! (

0



1)+ (

e

1



2) + (

e

1+

e

2



1)



(

0



2) 7! (

0



1)+ (

e

1



2)



E

1( ) : ;(;

e

1



1 ) 7! ;(;

e

1



1 ) ;(;

e

1+

e

2



1 ) ;(;

e

1+

e

2



2 )



(;

e

2



2 ) 7! (;

e

2



2 ) + ( ;

e

2



1 )

:

These are displayed in gure 6. It is interesting to remark that the image, by the dual substi-tution

E

1( ), of each unit tip is a connected path. This is in fact characteristic, among unitary substitutions on 2 letters, of the so called invertible substitutions (see Wen-Wen]), also called sturmian substitutions since their xed point is a sturmian sequence. The next example shows what happens in the non-invertible case it would be interesting to nd analogous properties for 3 or more letters.

Note also that the connectivity property for the dual map arises only if we consider the geometric representation linked to the duality morphisms

'

k if we do not shift the point

x

by

e

i1+

+

e

i

k, many convenient properties, in this and other cases, are lost.

Example

2

.

Let be the substitution

: 1 !112 2 !21

:

The matrix

A

is the same as the one for Fibonacci substitution given in Example 1. But this is not invertible substitution. It is easy to check this fact by using EI's Theorem. Then the linear

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-(

0



1) E1 ( ) 7;! (

0



2) -6 -6 E1 ( ) 7;! -6 6 ;(;

e

1



1 ) E1 ( ) 7;! (;

e

2



2 ) 6 6  -E1 ( ) 7;! -?

Figure 5. The gure of

E

1( ) and

E

1( ) in Example 1 -(

0



1) E1 ( ) 7;! (

0



2) - -6 6 E1 ( ) 7;! 6 -6 ;(;

e

1



1 ) E1 ( ) 7;! (;

e

2



2 ) 6  6 -E1 ( ) 7;! ?

-Figure 6. The gure of

E

1( ) and

E

1( ) in Example 2 mapping

E

1( ) and the dual mapping

E

1( ) are given by

E

1( ) : (

0



1) 7!(

0



1)+ (

e

1



1) + (2

e

1



2)



(

0



2) 7!(

0



2)+ (

e

2



1)



E

1( ) : ;(;

e

1



1 ) 7!;(;

e

1+

e

2



1 ) ;(;2

e

1+ 2

e

2



1 ) ;(;

e

2



2 )



(;

e

2



2 ) 7!(;

e

1



1 ) + (

e

1 ;2

e

2



2 )

:

These are displayed in gure 6.

Example

3

.

Let be the substitution

: 1 !23 2 !123 3 !1122233

:

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; ;  E1 ( ) 7;! -6 ; ; (

0



1) -; ; (

0



2) E1 ( ) 7;! ; ;  -6 ; ; 6 (

0



3) E1 ( ) 7;! ; ;  ; ; - - -6 6 ; ; ; ; E2 ( ) 7;! ; ; ; ; ; ; (

0



1^2) ; ; ; ; (

0



1^3) E2 ( ) 7;! ; ; ; ; ; ; ; ; ; ; ; ; ; (

0



2^3) E2 ( ) 7;! ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; (

0



1^2^3) E3 ( ) 7;! ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

Figure 7. The gure of

E

k( ),

k

= 1



2



3 in Example 3

This substitution is an example which is unimodular but not an invertible endomorphism on the free group of rank 3. The matrix

A

and the inverse matrix

A

;1 are

A

= 2 4 0 1 2 1 1 3 1 1 2 3 5

and

A

;1 = 2 4 ;1 0 1 1 ;2 2 0 1 ;1 3 5

:

Then the linear mappings

E

k( ),

k

= 1



2



3, are given by

E

1( ) : (

0



1) 7!(

0



2)+ (

e

2



3)



(

0



2) 7!(

0



1)+ (

e

1



2) + (

e

1+

e

2



3)



(

0



3) 7!(

0



1)+ (

e

1



1) + (2

e

1



2) + (2

e

1+

e

2



2) +(2

e

1+ 2

e

2



2)+ (2

e

1+ 3

e

2



3) + (2

e

1+ 3

e

2+

e

3



3)



E

2( ) : (

0



1 ^2) 7!;(

0



1^2);(

e

2



1 ^3)



(

0



1^3) 7!;(

0



1^2);(

e

2



1 ^3);(

e

1



1 ^2);(

e

1+

e

2



1 ^3) ;(2

e

1+

e

2



2 ^3);(2

e

1+ 2

e

2



2 ^3) + (2

e

1+ 3

e

2+

e

3



2 ^3)



(

0



2^3) 7!;(

e

1



1 ^2);(

e

1+

e

2



1 ^3) + (2

e

1+

e

2



1 ^2);(2

e

1+

e

2



1 ^3) +(2

e

1+ 2

e

2



1 ^2) + (2

e

1+ 3

e

2



1 ^3) + (2

e

1+ 3

e

2+

e

3



1 ^3) ;(3

e

1+

e

2



2 ^3);(3

e

1+ 2

e

2



2 ^3) + (3

e

1+ 3

e

2+

e

3



2 ^3)



E

3( ) : (

0



1 ^2^3) 7!(2

e

1+

e

2



1 ^2^3) + (2

e

1+ 2

e

2



1 ^2^3) ;(2

e

1+ 3

e

2+

e

3



1 ^2^3)

:

These are displayed in gure 6. Observing this gure, where we can recognize that the image of the unit cube at the origin consists in 3 cubes (with dierent orientations), we can make the conjecture that the necessary condition of Proposition 5.2 is also sucient.

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Example

4

.

Let be the substitution

: 1 !12 2 !13 3 !14 4 !1

:

This substitution can be considered as the natural generalization, on 4 letters, of the Rauzy substitution. One can easily compute the extensions and dual extensions. This is a case where results of the last part of section 4 apply, since it is a unimodular Pisot substitution satisfying the coincidence condition.

The pictures are not easy to draw, since we are in dimension 4 however, it is possible to show the limit set on the stable space, which looks a bit like a potato. The exact gure can be found as the domain of the Potato exchange transformation in Ito-Miz].

7.

Additional remarks

A rst remark is that a large part can be immediately generalized from substitutions to endomorphisms of free groups. The main interest of considering substitutions is that, in this case, there is no cancellation this makes it easy, in particular, to study

E

1( ). However, this reason disappears for higher dimensional extension.

To generalize this framework to the free group, one must be able to de ne the element (

x

i

;1). Since we want the path associated to the word

w

=

ii

;1 to be empty, the natural solution is to de ne (

x

i

;1) as

;(

x

;

e

i

i

). The rest follows easily from this de nition.

A second remark is that there is certainly an underlying homological theory, which would make all these contructions natural the maps

'

k de ned in section 1 seem to be a kind of

Poincare duality.

It is also possible to make the union of the Gk into a graded module, by de ning an exterior product. It is given onG

1 by (

x

i

)

^(

y

j

) = (

x

+

y

i

^

j

), and the generalization to the other sets is straightforward. It is interesting to note that this leads to a simple de nition of

E

2( ), as

E

1( )

^

E

1( ).

A last remark is that, instead of considering one substitution and its powers, we can consider the products of a sequence of substitutions this is a non-commutative generalization of the product of a sequence of matrices. In this setting, we can build the extensions of these prod-ucts this can allow, for example, to build explicitly discrete approximations of a plane, using generalized continued fractions, as is done in the paper Arn-Ber-Ito].

References

Arn-Ber-Ito] P.Arnoux, V. Berthe, Sh.Ito,Discrete planes, Z 2

-actions, Jacobi-Perron algorithm and substitu-tions, preprint.

Arn-Ito] P.Arnoux, Sh.Ito,Pisot Substitutions and Rauzy Fractals, Pretirage IML98-18, preprint

submit-ted.

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Dek2] F.M.Dekking, replicating supergures and endomorphisms of free groups, J. Combin. Theory Ser. A32(1982), 315{320.

Ei-Ito] H.Ei, Sh.Ito,Decomposition Theorem on Invertible Substitutions, Osaka J.Math.35(1998),821-834.

Ito-Ohtsuki] Sh.Ito, M.Ohtsuki, Modied Jacobi-Perron Algorithm and Generating Markov Partitions for Spe-cial Hyperbolic Toral Automorphisms, Tokyo J. Math.16(1993), 441-472.

Ito-Kimura] Sh.Ito, M.Kimura,On Rauzy fractal, Japan J. Indust. Appl. Math.8(1991), 461-486.

Ito-Miz] Sh.Ito, M.Mizutani,Potato Exchange Transformations with Fractal Domains, preprint.

MaKaSo] W.Magnus, A.Karrass, D.Solitar,Combinatorial group theory, Wiley Interscience, New York 1966. Messaoudi] A.Messaoudi,Autour du Fractal de Rauzy, These, Universite d'Aix-Marseille2(1996).

Rauzy] G.Rauzy,Nombres algebriques et substitutions, Bull. Soc. Math. France110(1982), 147-178.

Wen-Wen] Z.-X. Wen, Z.-Y. Wen,Local Isomorphisms of Invertible Substitutions, C. R. Acad. Sci. Paris,318

Serie I (1994), 299-304.

(Yuki Sano) Department of Mathematics, Tsuda College, Tsuda-Machi, Kodaira, Tokyo 187, Japan

E-mail address:sano@tsuda.ac.jp

(Pierre Arnoux) Institut de Mathematiques de Luminy (UPR 9016), 163 Avenue de Luminy, Case 930, 13288 Marseille Cedex 9, France

E-mail address:arnoux@iml.univ-mrs.fr

(Shunji Ito)Department of Mathematics, Tsuda College, Tsuda-Machi, Kodaira, Tokyo 187, Japan

Figure

Figure 1. The map E 1 ( ) for Rauzy substitution
Figure 3. The  gure of E k ( ), k = 1  2 and the commutation with boundary for Rauzy substitution
Figure 4. The  gures of E k ( ), k = 1  2 and the commutation with coboundary for Rauzy substitution
Figure 6. The  gure of E 1 ( ) and E 1 ( ) in Example 2 mapping E 1 ( ) and the dual mapping E 1 ( ) are given by
+2

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