## Generalized substitutions, tilings and numeration June 11-13, 2007, Porquerolles (France) Program

### Escaping from the substitution Pisot case

• 10.00 - 11. 00 Arrival of participants
• 11.00 - 11.25 Maki Furukado
Tilings generated by non-Pisot companion matrices
• 11.30 - 11.55  Philippe Narbel
Linear complexity languages and laminations
• 12.00 - 12.25 Hiromi Ei
Generalized Rauzy fractals generated by automorphisms of the free group with two letters
• 12.30 - 12.55 Bernd Sing
Non-unimodular Pisot substitutions

### Higher dimensional continued fraction.

• 9.00 - 9.25 Shin-ichi Yasutomi
Jacobi-Perron Algorithm and Diophantine Approximation to cubic numbers
• 9.30- 9.55 Thomas Fernique
Free group morphisms, Brun algorithm and digital plane recognition
• 10.00 - 10.25 Jun-Ichi Tamura
Playing with new algorithms of continued fractions of higher dimension.

### Number Systems and tilings

• 11.00 - 11.25 Julien Bernat
Factorization and primality on beta-integers
• 11.30 - 11.55 Joerg Thuswaldner
Number systems and tilings over Laurent series
•  12.00 - 12.25 Dirk Freottloeh
• 12.30 - 12.55 Fumihiko Enomoto
Tilings of a Riemann surface and cubic Pisot numbers

### Dynamical systems and numeration

• 9.00 - 9.25 Shigeki Akiyama
• 9.30 - 9.55 Ali Messaoudi
• 10.00 - 10.25 Koshiro Ishimura
Characterization of periodic points of the negative slope
algorithm
• 11.00 - 11.25 Jean-Francois Bertazzon
Piecewise isometries in the plane
• 11.30 - 11.55 Yann Jullian
Substitutions on trees
• 12.00 - 12.25 Mathieu Sablik

## Abstracts

### Shigeki Akiyama Pentagonal packing and S-adic system

Dynamical systems associated to aperiodic orbits of the domain exchange associated to $2\pi/5$ rotation is discussed. We associate two numeration systems, additive and multiplicative coding to this system
and study its induced systems.

### Julien Bernat Factorization and primality on beta-integers

The set of Beta-integers does not satisfy standard 'good' properties for performing arithmetics. We discuss here a construction that allows us to define and study the notions of factorization and primality in this framework

### Jean-Francois Bertazzon Piecewise isometries in the plane

We study a class of piecewise isometry of the euclidian plane. When not bijective, they are either globbally attractive or repulsive.  The bijective case is more delicate and gives rise to "fractal" tilings of the plane. It was proved by Quas and Goetz that in this case the maps are recurent. We will explain this result using a dynamical induction point of view.

### Hiromi Ei Generalized Rauzy fractals generated by automorphisms of the free group with two letters

In recent researches, we begin to discuss the case without a real Pisot condition (cf.[3]). In such case, automorphisms are used instead of substitutions. For a Pisot, unimodular substitution, Rauzy fractals are constructed as a realization of the symbolic dynamical system related to the fixed point of a substitution, and we want to keep the construction for automorphisms. In the paper [1], the method to construct Rauzy fractals associated with an automorphism with some condition is proposed, using a double substitution.
In my talk, we consider automorphisms which are conjugate to invertible substitutions through an example of automorphism. And we construct the generalized Rauzy fractals and the domain exchange which is isomorphic
to the rotation on the two dimensional torus.

### FumihikoEnomoto Tilings of a Riemann surface and cubic Pisot numbers

Using the reducible algebraic polynomial x5 - x4-1 = ( x2 -x + 1)( x3 - x - 1 ), we study two types of tiling substitutions  $\tau^*$ and $\sigma^*$: $\tau^*$ generates a tiling of a plane based on five prototiles of polygons, and $\sigma^*$ generates a tiling of a Riemann surface, which consists of two copies of the plane, based on ten prototiles of parallelograms. Finally we claim that $\tau^*$-tiling of $\mathcal{P}$ equals a re-tiling of $\sigma^*$-tiling of $\mathcal{R}$ through the canonical projection of the  Riemann surface to the plane.

### Thomas Fernique Free group morphisms, Brun algorithm and digital plane recognition

Dual maps of substitutions (that is, non erasing morphisms of the free monoid) have been introduced around 20 years ago (P. Arnoux, S. Ito). Those are maps defined over (d-1)-dim. faces of unit hypercubes of R^d. Some extensions have been provided : dual maps acting over lower dim. faces (P. A., S. I., Y. Sano, M. Furukado) and dual maps of free group morphisms (H. Ei).
Brun algorithm is used to define Brun expansions of real vectors, which can be seen as a (particular) multi-dimensional extension of continued fraction expansions.
Last, stepped hyperplanes are sets of faces of unit cubes of R^d which digitalize real hyperplanes. Stepped hypersurfaces extend this definition to any set of faces which is homeomorphic to a real hyperplane.
In this talk, we show how the dual maps of particular free group morphisms allows, given a stepped hyperplane P, to compute a sequence of stepped hyperplanes (P_n), such that the Brun algorithm applied to the normal vector of P exactly yields the sequence of normal vectors of the P_n's. In other words, we use dual maps for defining Brun expansions of stepped hyperplanes.
The advantage of this way of defining Brun expansions of stepped  hyperplanes is that this can be easily extended to stepped hypersurfaces. In particular, we show how it is connected with digital plane recognition, that is, how it can be decided whether a given stepped hypersurface is a stepped hyperplane or not.
Numerous nice graphical examples should help to increase span attention (I hope).

### Dirk Freottloeh About duality of cut-and-project sequences

In connection with self-similar cut-and-project tilings, or model sets, or Sturmian sequences, several notions of 'duality' occur. This talk aims to giving an overview over the different concepts. With distinguishing the cases of one-dimensional tilings and d-dimensional tilings, we show connections between the different concepts. These include 'Galois-dual tilings' (Thurston, Gelbrich,...), tilings arising from the 'natural decomposition' of the window (Sirvent, Wang,...), 'dual substitutions' (Arnoux, S. Ito,...),  'invertible substitutions' (Wen-Wen, Ei,...) and
more (Harriss,...).

### Maki Furukado Tilings generated by non-Pisot companion matrices

Starting from the unimodular Pisot matrix A $\in$  GL(d, $\mathbb{Z}$), there are many articles about how we obtain the polygonal/self-affine quasi-periodic tilings on the (d-1)-dimensional A-contracting invariant plane. In this talk, we will give the polygonal/self-affine quasi-periodic tilings on the 2-dimensional A-contracting invariant plane from the non-Pisot hyperbolic matrix A $\in$ GL(4,$\mathbb{Z}$) called the companion matrix.

### Koshiro Ishimura Characterization of periodic points of the negative slope algorithm

S.Ferenczi, C.Holton and L.Zamboni introduce the negative slope algorithm in the relation between three-interval exchange transformations and three-letters languages. They show the necessary and sufficient condition for eventually periodicity of the negative slope algorithm. We show the necessary and sufficient condition for purely
periodicity of the negative slope algorithm by using the natural extension method.

### Yann JullianSubstitutions on trees

We define a notion of coloured tree and substitutions acting on these objects. We explore the notion of fixed points of such "substitutions" having in perspective a study of dynamics on these trees.

### Philippe Narbel Linear complexity languages and laminations

Surface laminations can be symbolically represented as languages over finite alphabets with exact linear complexity. With this respect, we shall show how to effectively construct families of them by using
substitution compositions, while pointing out several open problems.

### Bernd Sing Non-unimodular Pisot substitutions

We will discuss non-unimodular Pisot substitutions. Using an example, we will show how the things known for unimodular substitutions look like in the unimodular case. These include the stepped surface, periodic and aperiodic tilings of the internal space, Markov partitions etc.

### Jun-Ichi Tamura Playing with new algorithms of continued fractions of higher dimension.

The speaker will report some phenomena in a series of experiments by a computer concerning continued fractions of higher dimension obtained by some old and new algorithms. Some of the phenomena will be curious, some are
interesting, and one of them seems to be profound. Not by a mathematical proof, but by the experiments, everybody will see the reason why the classical conjecture related to the periodicity of the continued fractions obtained by Jacobi-Perron algorithm seems to be false. On the other hand, I hope, everybody can believe that some of the new algorithms have nice properties related to the periodicity.

### Joerg Thuswaldner Number systems and tilings over Laurent series

Let $\F$ be a field and $\F[x,y]$ the ring of polynomials in two variables over $\F$. Let $f \in \F[x,y]$ and consider the residue class ring $R := \F[x,y]/f \F[x,y]$. Our first aim is to study digit representations in $R$, i.e., we ask for which $f$ each element $r \in R$ admits a digit representation of the form $d_0 +d_1 x + \cdots + d_\ell x^\ell$ with digits $d_i \in \F[y]$ satisfying $\deg_y d_i < \deg_y f$. These digit systems are motivated by the well-known notion of canonical number system.  In a next step we enlarge the space of representations in order to get representations with respect to negative powers of thebase'' $x$. It turns out that the appropriate spaces for such representations are $\F(x)[y] / f\F(x)[y]$ and $\F((x^{-1}, y^{-1})) / f \F((x^{-1}, y^{-1}))$, respectively. We characterize digit representations in these spaces and give easy to handle criteria for finiteness and periodicity. Finally, we attach fundamental domains to our number systems. The fundamental domain of a number system is the set of all numbers having only negative powers of $x$ in their $x$-ary'' representation. Interestingly,the fundamental domains of our number systems set turn out to be a unions of boxes. If we choose $\F=\F_q$ to be a finite field,these unions become finite.

### Shin-ichi Yasutomi Jacobi-Perron Algorithm and Diophantine Approximation to cubic numbers

For periodic (x,y) with Jacobi-Perron Algorithm, we show some properties about Diophantine Approximation to (x,y).