June 11-13, 2007, Porquerolles (France)

Program

- 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*

- 9.00 - 9.25
**Shin-ichi Yasutom****i***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.*

- 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* **Dirk Freottloeh***About duality of cut-and-project sequences*- 12.30 - 12.55
**Fumihiko****Enomoto***Tilings of a Riemann surface and cubic Pisot numbers*

- 9.00 - 9.25
**Shigeki Akiyama***Pentagonal packing and S-adic system* - 9.30 - 9.55
**Ali Messaoudi***Eigenvalues of exotic adding machines,* - 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**

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.

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

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.

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.

Using the reducible algebraic polynomial
x^{5} - x^{4}-1 = ( x^{2} -x + 1)( x^{3} - 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.

*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).

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,...).

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.

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.

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.

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.

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.

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.

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 the``base'' $x$. 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.

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