Symbiose Project Team - INRIA/Irisa © 2007 - 2008

 Rauzy Fractals
Written by Anne SIEGEL

## Paper abstract on numeration, symbolic dynamics, tilings and Rauzy fractals (A. Siegel)

Applying substitutions to understand the connection between tilings, ergodic theory, discrete geometry and non-integer numeration systems.
An overview on Rauzy fractal.

## Overview

Substitutions or iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters with words. It naturally generates a minimal symbolic dynamical system that have many arithmetical, geometrical and dynamical properties. The Fibonacci morphism $\sigma(1)=12$, $\sigma(2)=1$ (related to the golden ratio) provided a good illustration of these different properties and the relations between them (see details in Bordeaux Summer school, with Arnoux).

In some specific case (unimodular morphism of Pisot type), iterated morphisms can be understood in a geometrical framework, thanks to the construction of a Rauzy fractal, that is, a self-similar compact subset of the Euclidean space. A fundamental question about dynamical systems associated with an iterated morphism is whether they have a pure discrete spectrum. A construction of Rauzy fractals by power series with digits governed by a graph (work with V. Canterini, Journal AMS) allows to give an algorithm and an effective necessary and sufficient condition to answer this question, by describing the intersection between the Rauzy fractal and its translated copies under a lattice tiling (Ann. Fourier).

From another point of view, Rauzy fractals may generate several classes of covering. Deciding wether the coverings are tilings is deeply related the question of pure discrete spectrum. In some specific case, these tilings appear to be equal to Thurston's tilings associated to a $\beta$-numeration system. The definition of beta-numeration systems can be extended to define substitution numeration system where the digits are no more integers but polynomials that describe the combinatorics of the substitution. Then, tilings generated by Rauzy fractals appear to be generalized Thurston's tilings. Then, conditions for tilings can be given by extending the (F) property introduced by Akiyama, Frougny and Solomyak (work with V. Berthe, Integers).

From the point of view of arithmetics, it is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with $10$. This result extends to beta-expansions with a Pisot base beta which is not necessarily a unit: real numbers having a purely periodic expansion in such a base are characterized thanks to Rauzy fractals (work with V. Berthe, J. Number Theory).

A complementary formalism to build Rauzy fractals lies in discrete geometry: Arnoux and Ito extend the definition of a substitution to faces in the space. By using this definition, a formalism can be obtain for two-dimensional iterated morphisms, that is, a substitution mapping ${\mathbb Z}^2$ to ${\mathbb Z}^2$ (work with Arnoux, Berthe, TCS). It is shown that they can be iterated by using local rules, and that they generate two-dimensional patterns related to discrete approximations of irrational planes with algebraic parameters. Such a two-dimensional iterated morphism can be associated with any usual Pisot unimodular one-dimensional iterated morphism over a three-letter alphabet. This construction allows one to provide discrete geometry conditions for tilings and pure discrete spectrum.

Therefore, Rauzy fractals appears as connected points between numeration systems, substitutive symbolic dynamical systems and discrete geometry. My previous researchs with V. Berthe allowed to state the correspondance between these different points of view. The best tools of each points of view can be used to answer questions in other domains. For instance

• Rauzy fractals can be define in the non unit case by using $p$-adic representations. These representations shall be used to study beta-numeration systems in the non unit case and obtain some results on the existence of points with purely periodic expansions (collaboration with Akiyama, Berthe and Barat).
• Rauzy fractals appear as fixed points of generalized Iterated Function Systems. However, much is known on simple connectivity of fixed point of IFS. With J. Thuswaldner (Austia), we provide explicit condition for simple connectivity or non trivial fundamental group of Rauzy Fractals.
• Objects that are used to construct Rauzy fractals are substitution, that is, morphism of the free group with no cancellation. Levitt and Lustig define the notion of complexity of morphisms of free groups in some specific cases. This could allow to extend the definition of Rauzy fractals to morphisms with a small number of cancellations (collaboration with Arnoux, Levitt, Lustig, Hillion).
• Discrete geometry is a good tool to define new conditions for tilings; this will allows to explore the generalization of Rauzy fractal to the combinaison of different substitutions.

## Rational numbers with purely periodic beta-expansion

B. Adamczewski, C. Frougny, A. Siegel, W. Steiner,  Bulletin of the London Mathematical Society.

We study real numbers beta with the curious property that the beta-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let gamma(beta) denote the supremum of the real numbers c in (0, 1) such that all positive rational numbers less than c have a purely periodic beta-expansion. We prove that gamma(beta) is irrational for a class of cubic Pisot units that contains the smallest Pisot number beta. This result is motivated by the observation of Akiyama and Scheicher that gamm(beta) = 0.666 666 666 086 · · · is surprisingly close to 2/3.

## Fractal tiles associated with shift radix systems

V. Berthé, A. Siegel, W. Steiner, P. Surer, J. Thuswaldner, submitted.

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.
In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the wellknown tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.
We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).

## Substitutions, Rauzy fractals and tilings

V. Berthé, A. Siegel, J. Thuswaldner  chapter 5 of  Combinatorics, Automata and Number Theory, Cambridge University Press, 2010

This chapter focuses on multiple tilings associated with substitutive dynamical systems. We recall that a substitutive dynamical system is a symbolic dynamical system where the shift S acts on the set Xσ of infinite words having the same language as a given infinite word which is generated by powers of a primitive substitution σ. We restrict to the case where the inflation factor of the substitution σ is a unit Pisot number. With such a substitution σ, we associate a multiple tiling composed of tiles which are given by the unique solution of a set equation expressed in terms of a graph associated with the substitution σ: these tiles are attractors of a graph-directed iterated function system (GIFS). They live in Rn−1, where n stands for the cardinality of the alphabet of the substitution. Each of these
tiles is compact, it is the closure of its interior, it has non-zero measure and it has a fractal boundary that is also an attractor of a GIFS. These tiles are called central tiles or Rauzy fractals, according to G. Rauzy  (Rauzy 1982).
Central tiles were first introduced in (Rauzy 1982) for the case of the Tribonacci substitution and then in (Thurston 1989) for the case of the beta-numeration associated with the Tribonacci number (which is the positive root of X3 − X2 − X − 1). One motivation for Rauzy’s construction was to exhibit explicit factors of the substitutive dynamical
system as translations on compact abelian groups, under the hypothesis that σ is a Pisot substitution. By extending the seminal construction in (Rauzy 1982), it has been proved that central tiles can be associated with Pisot substitutions (see e.g. (Arnoux and Ito 2001) or (Canterini and Siegel 2001b)) as well as with beta-numeration with respect to Pisot numbers (cf. (Thurston 1989), (Akiyama 1999) and (Akiyama 2002)). They are conjectured to induce
tilings in all these cases. The tiling property is known to be equivalent to the fact that the substitutive dynamical system has pure discrete spectrum (see (Pytheas Fogg 2002, Chapter 7) and (Barge and Kwapisz 2006)) when σ is a unit Pisot irreducible substitution. We have chosen here to concentrate on tilings associated with substitutions for the sake of clarity. A similar study can be performed in the framework of beta-numeration, with both viewpoints being intimately connected through the notion of beta-substitution. Indeed, a beta-substitution can be associated with any Parry number β (for more details, see Exercise 5.1 and Section 5.11). In the case where β is a Pisot number, the associated substitution can be Pisot reducible as well as Pisot irreducible. The exposition of the theory of central tiles is much simpler when σ is assumed to be Pisot irreducible, even if it extends to the Pisot reducible case. Hence, we will restrict ourselves to the Pisot irreducible case.
There are several approaches for the definition of central tiles. We detail below a construction for unit Pisot substitutions based on a broken line which is defined in terms of the abelianisation of an infinite word generated
by σ. Projecting the vertices of this broken line to the contractive subspace of the incidence matrix of σ along its expanding direction and taking the closure of this set yields the central tile. For more details on different approaches, see the surveys in (Pytheas Fogg 2002, Chapters 7 and 8) and (Berth´e and Siegel 2005), as well as the discussion in (Barge and Kwapisz 2006) and (Ito and Rao 2006). The aim of this chapter is to list a great variety of tiling conditions, by focusing on effectivity issues. These conditions rely on the use of various graphs associated with the substitution σ.

We first define the central tile as well as its decomposition into subtiles. Two (multiple) tilings associated with σ are then introduced in Section 5.3. The first one is called tiling of the expanding line. The second one is a priori not a tiling, but a multiple tiling. It is made of translated copies of the subtiles of the central tile. It is called the self-replicating multiple tiling. Note that it is conjectured to be a tiling. It will be the main objective of the present chapter to introduce various graphs that provide conditions for this multiple tiling to be a tiling. The first series of tiling conditions is expressed in geometric terms directly related to properties of the self-replicating multiple tiling. We start with a sufficient tiling property inspired by the so-called finiteness property (F). This leads us to introduce successively
several graphs, yielding necessary and sufficient conditions. We then discuss further formulations for the tiling property expressed in terms of the tiling of the expanding line. They can be considered as dual to the former set of
conditions. In particular, a formulation in terms of the so-called overlap coincidence condition is provided, as well as a further effective condition based on the notion of balanced pairs.

## Topological properties of Rauzy fractals

Anne Siegel and Joerg Thuswaldner, Mémoires de la SMF, en révision

Substitutions are  combinatorial objects (one replaces a  letter by a word) which  produce sequences  by iteration.  They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, {\it i.e.}, we study properties of the {\it Rauzy fractals} associated to substitutions.
To be more precise, let $\sigma$ be  substitution over the alphabet $\mathcal{A}$. We assume that the linearized matrix of $\sigma$ is primitive and its dominant eigenvalue is a unit Pisot number ({\it i.e.}, an algebraic number whose norm is one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to $\sigma$ a set $\T$
which is now called {\it central tile} or {\it Rauzy fractal} of $\sigma$. Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in $n=\#\mathcal{A}$ subtiles $\T(1),\ldots ,\T(n)$. The central tile
as well as its subtiles are graph directed self-affine sets that often have fractal boundary.
Pisot substitutions and central tiles are naturally of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix
representation. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and geometry of the underlying central tile.
After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of $\T$ and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether
$\T$ and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a disk and derive properties of their fundamental group.
The basic tools for our criteria are several classes of graphs built from the description of the tiles $\T(i)$ ($1\le i\le n$) as graph directed iterated function systems and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. They can be used to construct the
boundaries $\partial\T$ as well as $\partial\T(i)$ ($1\le i\le n$) and all points where two, three or four different tiles of the mentioned tilings meet.
When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for everyday's life'' when dealing with
topological and geometric properties of substitutions.
Many examples to illustrate our results are given. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

## Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

S. Akiyama, G. Barat, V. Berthé, A. Siegel Monashefte für Mathematik, 2008

This paper studies tilings and representation sapces related to the beta-transformation when beta is a Pisot number (that is not supposed to be a unit). The obtained results are applied to study the set of rational numbers having a purely periodic beta-expansion. We indeed make use of the connection between pure periodicity and a compact self-similar representation of numbers having no fractional part in their beta-expansion, called central tile: for elements x of the ring Z[1/beta], so-called x-tiles are introduced, so the central tile is a finite union of x-tiles up to translation. These x-tiles provide a covering (and even in some cases a tiling) of the space we are working in. This space, called complete representation space, is based on Archimedean as well as non-Archimedean completions of the number field Q(beta) for primes dividing the norm of beta. This representation space has numerous potential implications. We focus here on one application concerning the gamma function (gamma(beta) defined as the supremum of the set of elements v in [0, 1] such that every positive rational number p/q, with p/q <= v and q coprime with the norm of beta, has a purely periodic beta-expansion. Our study relies on the description of the topological properties of central tiles in terms of boundary graphs. Special focus is given to some quadratic examples, showing that the behaviour of gamma(beta) in the non-unit case is slightly different from its behaviour in the unit case.

## Fractal representation of the attractive lamination of an automorphism of the free group

P. Arnoux, V. Berthe, A. Hilion and A. Siegel, Annales de l'Institut Fourier 56(7), 2006, 2161-2212

In this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, the so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers ({\it iwip}) automorphisms) is given in the case where the dilation coefficient of the automophism is a unit Pisot number. The flow on the lamination, when discretized as a shift map in a symbolic dynamical system, is proved, in this case, to be measure-theoretically isomorphic to a domain exchange on a self-similar Euclidean compact set. This set is called the central tile of the automorphism, and is inspired by Rauzy fractals associated to Pisot primitive substitutions. The central tile admits some specific symmetries, and is conjectured under the Pisot hypothesis to be a fundamental domain for a toral translation.

## Tilings associated with beta-numeration and substitutions

V. Berthe and A. Siegel , INTEGERS (Electronic Journal of Combinatorial Number Theory), 5(3), pp. A2, 2005

This paper surveys different constructions and properties of some multiple tilings (that is, finite-to-one covering s) of the space that can be associated with beta-numeration and substitutions. It is indeed possible, generalizing Rauzy's and Thurston's constructions, to associate in a natural way either with a Pisot number $\beta$ (of degree $d$) or with a Pisot substitution $\sigma$ (on $d$ letters) some compact basic tiles that are the closure of their interior, that have non-zero measure and a fractal boundary; they are attractors of some graph-directed Iterated Funtion System. We know that some translates of these prototiles under a Delone set $\Gamma$ (provided by $\beta$ or $\sigma$) cover $\GR ^{d-1}$; it is conjectured that this multiple tiling is indeed a tiling (which might be either periodic or self-replicating according to the translation set $\Gamma$). This conjecture is known as the Pisot conjecture and can also be reformulated in spectral terms: the associated dynamical systems have pure discrete spectrum. We detail here the known constructions for these tilings, their main properties, some applications, and focus on some equivalent formulations of the Pisot conjecture, in the theory of quasicrystals for instance. We state in particular for Pisot substitutions a finiteness property analogous to the well-known (F) property in beta-numeration, which is a sufficient condition to get a tiling.

## Dynamique du nombre d'or

P. Arnoux et A. Siegel, Université d'été Sciences Mathématiques et Modélisation, Bordeaux, 2004

Un cours sur le nombre d'or et toutes ses facettes: geometrie, ecriture des entiers et des reels en base de Fibonacci, addition du nombre d'or et substitutions, multiplication par le nombre d'or, quasi-cristaux, generalisations.

## Purely Periodic beta-Expansions in the Pisot Non-unit Case

V. Berthe and A. Siegel Journal of Number Theory 153 (2) 2007, 153-172

It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with $10$. The aim of this paper is to extend this result to beta-expansions wi th a Pisot base beta which is not necessarily a unit: we characterize real numbers having a purely periodic expansion in such a base; this characterization is given in terms of an explicit set, called generalized Rauzy fracta l, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and $p$-adic spaces.

## Spectral theory for dynamical systems arising from substitutions

A. Siegel, EWM, K. Dajani, J. Von Reis (eds.), CWITract, Marseille, 2003.

Symbolic dynamical systems were first introduced to better understand the dynamics of geometric maps; particularly to study dynamical systems for which past and future are disjoint as for instance toral automorphisms or Pseudo-Anosov diffeomorphisms of surfaces. Self-similar systems are defined to be topologically conjugate to their own first return map on a given subset. A basic idea is that, as soon as self-similarity appears, a substitution is hidden behind the original dynamical system. In this lecture, we first illustrate this idea with concrete examples, and then, try to understand when symbolic codings provide a good representation. A natural queston finally arises: which substitutive dynamical systems are isomorphic to a rotation on a compact group? Partial answers have been given by many authors since the early 60's. Then, we will see how a spectral analysis problem finally reduces to a combinatorial problem, whose partial answers imply Euclidean geometry and even some arithmetics.

## Pure discrete spectrum dynamical system and periodic tiling associated with a substitution

A. Siegel, Annales de l'Institut Fourier 2(54), 2004, p. 288-299.

We give a computable sufficient condition for the symbolic dynamical system associated with a substitution of Pisot type to have a pure discrete spectrum. In the unimodular case, t his condition is necessary when the substitution has no nontrivial coboundary; it is satisfied if and only if the Rauzy fractal associated with the substitution generates a self-similar periodic tiling. The techniques used here also provide a computable sufficient condition for connectivity of the Rauzy fractal of a unimod ular substitution.

## Two-dimensional iterated morphisms and discrete planes

P. Arnoux, V. Berthe and A. Siegel, Theoretical Computer Science 319, 2004, p. 145--176.

Iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters by words. The aim of this paper is to introduce a formalism for a notion of two-dimensional morphisms; we show that they can be iterated by using local rules, and that they generate two-dimensional patterns related to discrete approximations of irrational planes with algebraic parameters. We associate such a two-dimensional morphism with any usual Pisot unimodular one-dimensional iterated morphism over a three-letter alphabet.

## Representation des systèmes dynamiques substitutifs non unimodulaires

A. Siegel, Ergodic Theory and Dynamical Systems, (2003), 23, 1247-1273.

On montre que la condition combinatoire de forte coïncidences est suffisante pour qu'un système substitutif de type Pisot soit isomorphe en mesure à un echange de morceaux dans un compact autosimilaire de mesure non nulle dans le produit d'un espace euclidien et d'extensions finies de corps $p$-adiques. Ainsi, tout système substitutif de type Pisot avec coïncidences est une extension finie de son facteur equicontinu maximal; en règle generale, ce dernier contient une translation $p$-adique si et seulement si la matrice d'incidence de la substitution est nilpotente modulo $p$.

## Substitutions in Dynamics, Arithmetics and Combinatorics

N. Pytheas-Fogg. Lectures Notes in Mathematics 1794, Springer-Verlag, 2002. Edited by V. Berthe, S. Ferenczi, C. Mauduit and A. Siegel.

A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure. The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a state-of-the-art survey of the more difficult and unsolved problems.

Keywords: Symbolic dynamics, automata sequences, combinatorics on words, ergodic theory, substitutive dynamical systems

## Automate des prefixes-suffixes associe à une substitution primitive

V. Canterini and A. Siegel, Journal de theorie des nombres de Bordeaux, 13(2), 2001, pp. 353--369.

On explicite une conjugaison en mesure entre le decalage sur un système substitutif minimal et une transformation adique sur un sous-shift de type fini, à savoir l'ensemble des chemins d'un automate dit des prefixes-suffixes. En caracterisant les preimages par la conjugaison des chemins periodiques de l'automate, on montre que cette conjugaison est injective sauf sur un ensemble denombrable, sur lequel elle est finie-à-un. On en deduit l'existence d'une suite de partitions du système qui est generatrice en mesure et une application aux fractals de Rauzy est donnee.

We prove that a substitutive dynamical system $\Omega$ is measurably conjugate to an adic transformation on a subshift of finite type defined as the set of paths on a graph. The conjugaison map is one-to-one except on the orbit of periodic points of $\Omega$, on which it is finite-to-one. We deduce a sequence of partitions of $\Omega$ which is is generating in measure and an application to Rauzy fractals is given.

## Geometric Representation of primitive substitution of Pisot type

V. Canterini and A. Siegel, Transactions of the AMS, 353(12), 2001, pp. 5121--5144.

We prove that a substitutive dynamical system of Pisot type contains a factor which is isomorphic to a minimal rotation on a torus. If the substitution is unimodular and satisfies a certain combinatorial condition, we prove that the dynamical system is measurably conjugate to an exchange of domains in a self-similar compact subset of the Euclidean space.

## Theorème des trois longueurs et suites sturmiennes : mots d'agencement des longueurs

A. Siegel, Acta Arithmetica, 97 (1-3), 2001, pp. 195--210.

Si $\alpha$ est irrationnel de convergents $\frac{p_n}{q_n}$, les points $\{k \alpha \}$ subdivisent pour ${0 \leq k < q_{n} + q_{n-1}}$, le cercle de perimètre $1$ en arcs ne prenant que deux longueurs. Le but de cet article est d'etudier l'ordre d'apparition de ces longueurs en les codant par un mot fini sur l'alphabet $\{0,1\}$. Nous montrons que ces mots sont equilibres et relions leurs valeurs d'adherence ultimement periodiques aux sous-suites de quotients partiels de $\alpha$ qui tendent vers l'infini. Nous prouvons aussi que les valeurs d'adherence sturmiennes, s'il en existe, sont des extensions gauches de mots sturmiens caracteristiques ayant pour angle une valeur d'adherence de la suite de nombres $\frac{q_n}{q_n + q_{n-1}}$. Si l'angle $\alpha$ est quadratique, elles sont points fixes de substitutions.

## Representations geometrique, combinatoire et arithmetique des systèmes substitutifs de type Pisot

A. Siegel, Thèse de Doctorat, Universite de la Mediterranee, 2000.

On cherche à savoir si les systèmes dynamiques symboliques engendres par une substitution de type Pisot sont des systèmes dynamiques standards ou totalement nouveaux, en determinant lesquels sont isomorphes à une translation sur un groupe compact.

On obtient une representation combinatoire des systèmes substitutifs primitifs en montrant que chacun est isomorphe à une transformation adique sur le support d'un sous-shift de type fini defini naturellement à partir de la substitution.

Cette representation combinatoire permet d'exhiber un système de numeration sous-jacent à la substitution et de representer geometriquement les substitutions par une translation sur un tore. Ainsi, tout système substitutif de type Pisot admet pour facteur topologique une translation torique.

Si la substitution est unimodulaire et verifie une certaine condition combinatoire, cette representation torique est la projection sur le tore d'une representation dans l'espace euclidien par un echange de morceaux dans un compact auto-similaire appele fractal de Rauzy de la substitution.

Les proprietes du système de numeration sous-jacent à la substitution permettent de donner une condition necessaire et suffisante effective en terme d'automates pour que le fractal de Rauzy de la substitution engendre un pavage regulier de l'espace euclidien. Ceci produit une condition suffisante pour qu'une système substitutif de type Pisot soit isomorphe à une translation torique.

Ceci permet de definir des partitions de Markov explicites pour une classe d'automorphismes du tore.

Au sujet des systèmes non unimodulaires, on montre qu'une addition sur le groupe des entiers $p$-adiques est facteur d'un système substitutif dont la matrice d'incidence a un polynôme caracteristique irreductible si et seulement si cette matrice est nilpotente modulo~$p$.

Ceci laisse entrevoir la possibilite d'obtenir des partitions de Markov explicites pour l'extension naturelle d'une classe d'endomorphismes du tore.