## Generalized substitutions, tilings and numeration March 6--10, 2006, Marseille (France) Program

Monday, March 6th

• 9.00 - 9.15 Arrival of participants
• 9.20 - 9.40 A. Hilion Symbolic lamination of a free group automophism
• 9.50 - 10.10 P. Arnoux Rauzy fractals for a class of free group automorphisms
• 10.20 - 10.40 H. Ei Weak sufficient condition for a substitution to be invertible
• 11.00 - 11.20 S. Ito Domain exchange transformations on Rauzy fractals and odometer transformations
• 11.30 - 11.50 M. Furukado Examples of self-similar tilings from non-Pisot unimodular matrices
• 12.00 - 12.20 T. Fernique Action of generalized substitutions on functional stepped surfaces

Tuesday, March 7th

• 9.00 - 9.50 M. Barge Geometric realization of Pisot spaces - the view from Montana
• 10.00 - 10.20 R.J. Fokkink Beta numeration and expansive dynamics
• 11.00 - 11.20 T. Kamae Numeration system for the (3/2)-expansion
• 11.30 - 11.50 B. Sing Pisot substitutions and adelic spaces
• 12.00 - 12.20 F. Enomoto Negative $\beta$-transformation and Tiling

Wednesday, March 8th

• 9.00 - 9.50 J.-M. Gambaudo Tilings and translation surfaces
• 10.00 - 10.20 E. Harriss Encyclopedia of Substitution Rules
• 11.00 - 11.20 D. Frettloeh Self-duality of Galois-dual tilings
• 11.30 - 11.50 F. Durand Systemes de numeration et pavages
• 12.00 - 12.20 R. Tijdeman Substitutions, abstract number systems and the space filling property

Thursday, March 9th

• 9.00 - 9.50 B. Adamczewski Diophantine properties for some abnormal numbers
• 10.00 - 10.20 S.-i. Yasutomi Dynamical system and Diophantine Approximation and substitution
• 11.00 - 11.20 V. Sirvent Spectra of recurrence dimension for dynamically defined subsets of Rauzy Fractals
• 11.30 - 11.50 T. Sadahiro Multiple points of tilings associated with Pisot numeration systems
• 12.00 - 12.20 J.-i. Tamura Some algorithms of multidimensional continued fractions, and Pell equations of higher degree

Friday, March 10th

• 9.00 - 9.20 S. Akiyama Periodicity of discrete rotation and domain exchange I
• 9.30 - 9.50 W. Steiner Periodicity of discrete rotations and domain exchange II
• 10.00 - 10.20J.-P. Gazeau Poisson formulas and diffraction spectra of weighted Delone sets on beta-integers and beta-lattices
• 11.00 - 11.20 J. Thuswaldner The fundamental group of the Sierpinski gasket
• 11.30 - 11.50 X. Bressaud Peano curves, Rauzy fractals and Levitt trees
• 12.00 - 12.20 A. Fisher Unique ergodicity and velocity for adic transformations and solenoids

Monday, March 6th

A. Hilion Symbolic lamination of a free group automophism
Invertible substitutions are very special cases of free groups automorphisms. According to Bestvina-Feighn-Handel, one can associate an attractive lamination to every "iwip" automorphism. I will give the definition of laminations in free groups, and explain how the attractive lamination of an iwip automorphism can be derived from a train-track representative of the automorphism. Such objects allow us (P. Arnoux, V. Berthé, A. Siegel) to construct a Rauzy fractal for a given class of free groups automorphisms.

P. Arnoux Rauzy fractals for a class of free group automorphisms
We show that, using the result of the previous lecture by A. Hilion, we can extend the classical results of construction of the Rauzy fractal from substitutions to a class of automorphisms of the free group, the so-called "irreducible with irreducible powers" automorphisms with a Pisot dilation coefficient; in the non- substitutive case, we obtain remarkable symmetry properties. The construction raises some intriguing questions (existence of a flow on the algebraic lamination associated to an IWIP automorphism).

H. Ei Week sufficient condition for a substitution to be invertible
Nielsen gave a way to calculate the inverse of an endomorphism on the free group (or a substitution) by the method of reducing word. On the other hand, we know that the property of invertibility is characterised by an extension and dual extension of an endomorphism. The aim of my talk is to recall these results and bring up problems related to invertible substitutions.

S. Ito Domain exchange transformations on Rauzy fractals and odometer transformations

M. Furukado Examples of self-similar tilings from non-Pisot unimodular matrices
We introduce the examples of self-similar tilings from $4 \times 4$ hyperbolic unimodular non-Pisot integer matrix. The difficulty is that the positive polygonal tile is mapped to the negative and positive polygonal ones according to the tiling substitution, though the positive polygonal tile is mapped to only positive polygonal ones in the Pisot case. To manage the negative polygonal tile, the "blocking" method is proposed in this talk.

T. Fernique Action of generalized substitutions on functional stepped surfaces
A functional stepped surface is a set of translated faces of the unit cube of R3 which is homeomorphic to the real plane x+y+z=0 by the orthogonal projection onto this plane. In particular, stepped planes are stepped surfaces. We will speak about the natural way the generalized substitutions act on functional stepped surfaces

Tuesday, March 7th

M. Barge Geometric realization of Pisot spaces - the view from Montana
I'll summarize everything we ( Bev Diamond, Jarek Kwapisz, and I ) know about geometric realization. Most of this can be found in the preprints: "Geometric theory of unimodular Pisot substitutions" and "Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to beta-shifts", available at http://www.math.montana.edu/~jarek/Papers.html ; and in "Proximality in Pisot tiling spaces", arXiv:math.DS/0509023v1 1 Sep 2005.

R.J. Fokkink beta numeration and expansive dynamics
Multiplication by $\beta$ induces a dynamical system on $\Q(\beta)$. If $\beta$ is an algebraic integer, then $\Q(\beta)$ is a number field and we may consider the induced transformation on all its algebraic completions. Of particular interest are those completions in which $\beta$ has absolute value unequal to 1. Let their product be $K$. The ring $\Z[\beta,\beta^{-1}]$ form a lattice $L$ in $K$. On $K/L$, which is a compact abelian group, multiplication by $\beta$ induces a hyperbolic transformation. The interesting completions of $\Q(\beta)$ correspond to the stable and unstable manifold of this hyperbolic transformation. Finding numeration systems can then be seen as finding fundamental domains for $L$ in $K$. This is all very nice, but the picture seems to get muddy if $\beta$ is not an algebraic integer anymore but only algebraic, such as $2/3$.

T. Kamae Numeration system for the (3/2)-expansion
We define the translation and the scaling transformation on the compactification \Omega of R with respect to the (3/2)-expansion. The translation is the R-additive action and the scaling is the G-multiplicative action with G={(3/2)^n} satisfying the comutation relation g(w+t)=w+gt, where w\in\Omega, t\inR, g\inG, and by w+t and gw, we mean the additive and multiplicative action to w. Moreover, the R-action is strictly ergodic having 0-topological entropy with the unique invariant probability measure \mu, and the g-action with g\inG and g\ne1 has |log g|-topological entropy attained uniquely by \mu. We do not know whether {T^n 1} is dense in [0,1) or not, where Tx=(3/2)x in [0,2/3), =(3/2)x-1 in [2/3,1]. But assuming that this is dense, we obtained the zeta-funcion of \Omega with respect to the G-action.

B. Sing Pisot substitutions and adelic spaces
In the first part of the talk, we will show how the substitution matrix of a Pisot substitution (over d symbols) can be interpreted as principle lattice transformation in an adele. Projecting to its contracting and expanding eigenspaces, we then recover in the second part of the talk the 1-dimensional Pisot substitution and its associated Rauzy fractal. We will end the talk with some remarks about non-unit Pisot substitutions and their pure pointedness

F. Enomoto Negative $\beta$-transformation and Tiling
We shall introduce a negative $\beta$-transformation $T_{\beta}~(\beta < -1)$, which naturally induce an AH-substitution .Under the assumption of Markov property of $T_{\beta}$,the Markov partition of a group automorphism given by $T_{\beta}$ can be constracted by atomic surfaces on the contractive subspace and intervals on the expanding line

Wednesday, March 8th

J.-M. Gambaudo Tilings and translation surfaces
Consider a tiling of the 2-dimensional Euclidean plane made with a finit number of polygons (up to translation) that meet full face to full face, then these polygons can tile a compact translation surface. We will discuss this result and its link with Z2$action on the Cantor set and statistical mechanics E. Harriss Encyclopedia of Substitution Rules With Dirk Frettlöh I have been developing a online encyclopedia of substitution rules. I will announce the site and show some of the examples, both famous and less known. D. Frettloeh Self-duality of Galois-dual tilings Thurston introduced a construction of the (2dim) 'Galois-dual' of a certain one-dimensional tiling. This construction is essentially the same as the computation of a Rauzy fractal. Gelbrich applied this construction to a few two-dimensional tilings. In this talk, the Galois-duals of the Penrose- and the Ammann-Beenker tilings are presented. Moreover, the question about self-dual tilings (with respect to this Galois-duality) is addressed. Some necessary, and some sufficient conditions for this are obtained and illustrated. F. Durand systemes de numeration et pavages Un ensemble d entiers$E$est dit$p$-reconnaissable si le langage forme par les ecritures en base$p$des elements de$E$est reconnaissable par un automate. En 1969 A. Cobham a montre que si$p$et$q$n ont pas de puissance commune alors :$E$est$p$-reconnaissable et$q$-reconnaissable si et seulement$E$est une reunion finie de progressions arithmetiques. En 1977 Semenov a generalise ce resultat a des sous-ensembles de$\NN^k$. Les objets manipules font apparaitre des pavages auto-similaires du plan (par des carres colores). Recemment Ormes-Radin-Sadun et Holton-Radin-Sadun ont obtenu ce type de resultat pour des pavages de type Penrose. Dans cet expose je proposerai une preuve dynamique du theoreme de Semenov et j expliquerai pourquoi avec la meme methode on peut envisager d obtenir le resultat de Holton-Radin-Sadun. R. Tijdeman Substitutions, abstract number systems and the space filling property Clemens Fuchs and I studied multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We showed for a large class of substitutions that the resulting word is the restriction of a linear function mod 1 and that it can be decided whether the resulting word is space filling or not. In the presentation the main ideas will be sketched and some examples will be given Thursday, March 9th B. Adamczewski Diophantine properties for some abnormal numbers Despite some speculations, the decimal expansion of algebraic irrational numbers are still rather mysterious. Actually, the picture is not really better regarding other natural expansions, such as the continued fraction expansion or the Rényi$\beta$-expansion. In this talk, I will survey some recent applications of the Schmidt Subspace Theorem in connexion with this topic. Most of the results I will discuss have been obtained in joint works with Yann Bugeaud. S.-i. Yasutomi Dynamical system and Diophantine Approximation and substitution Dynamical systems related to substitutions give rich appication to Diophantine Approximatiion. In this context we show some topics from simultaneous Diophantine approximation and inhomogeneous Diophantine approximation. V. Sirvent Spectra of recurrence dimension for dynamically defined subsets of Rauzy Fractals We compute the spectra of the recurrence dimension for dynamically defined subsets of Rauzy fractals, in the case when these sets are totally disconnected. These subsets of the Rauzy fractals are defined by subadic systems, therefore there is a well defined dynamical system defined on them. First we compute the spectra of the recurrence dimension for adic and subadic systems and later we extend those results to the dynamical systems mentioned before, which are geometrical realizations of these symbolic systems. This dimension is characterized by the Poincar\'e recurrence of the system, and the corresponding spectrum is invariant under bi-Lipschitz transformations. We also the address the question, when the dynamically defined subsets of Rauzy Fractals are totally disconnected. T. Sadahiro Multiple points of tilings associated with Pisot numeration systems In this talk, we deal with a kind of aperiodic tilings associated with Pisot numeration systems, originally due to W. P. Thurston, in the formulation of S. Akiyama. We treat tilings whose generating Pisot units$\beta$are cubic and not totally real. % %Our main concern in this paper is to describe multiple %points of the tilings in cubic and not totally real %case and show the explicit description of the set of %triple points as a collection of model sets. % Each such tiling gives a numeration system on the complex plane; %Given such a tiling generated by a Pisot unit$\beta$, we can express each complex number$z$in the following form$:$$z=c_k\alpha^{-k}+c_{k-1}\alpha^{-k+1}+\cdots+c_1\alpha^{-1} +c_0 + c_{-1}\alpha1 + c_{-2}\alpha2 + \cdots %z=\sum_{k=k_0}^\infty c_{-k}\alpha^k$ where$\alpha$is a conjugate of$\beta$, and$c_{-m}c_{-m+1}\cdots c_{k-1}c_{k}$is the$\beta$-expansion of some real number for any integer$m\$. %We study these numeration systems representing complex numbers. %Specifically, We determine the set of complex numbers which have three or more representations. This is equivalent to determining the triple points of the tiling, which is shown to be a collection of model sets (or cut-and-project sets). We also determine the set of complex numbers with eventually periodic representations.

J.-i. Tamura Some algorithms of multidimensional continued fractions, and Pell equations of higher degree
We give something more about a system of fractionals and multidimensional continued fractions. In particular, we give some results related to Pell equations of higher degree coming not only from pure extensions but also from bi-quadratic extensions of the rational number field, etc.

Friday, March 10th

S. Akiyama Periodicity of discrete rotation and domain exchange I
We discuss periodicity of discretized rotations whose angles are rational to \pi which arose from shift radix systems. One can associate a domain exchange which has self similar structure. This is a joint work with H. Brunotte, A. Pethoe and W. Steiner.

W. Steiner Periodicity of discrete rotations and domain exchange II
Continuing Shigeki Akiyama's talk, we will discuss a substitution induced by the domain exchange for rotations corresponding to quadratic numbers. By using this substituion, it is possible to calculate all possible values of the period lengths

J.-P. Gazeau Poisson formulas and diffraction spectra of weighted Delone sets on beta-integers and beta-lattices
We examine the Fourier transform of a weighted Dirac comb of beta-integers within the framework of the theory of distributions. This transform can be well characterized when beta is a quadratic unit Pisot. and the result is used to compute the intensity function of the pure point part of the diffraction spectrum. The method is extended to the computation of diffraction spectra of weighted Delone sets on beta-lattices when beta is a cyclotomic quadratic Pisot number. We then examine the problem of extending these results to other numbers beta, like quadratic non-unit Pisot.

J. Thuswaldner The fundamental group of the Sierpinski gasket
We give a short characterization of the fundamental group of the Sierpinski gasket as a certain subset of a projective limit of groups. The projective limit is the Czech homotopy group of the gasket. The groups forming this projective limit are fundamental groups of natural approximations of the Sierpinski gasket. In future, we hope to find similar characterizations for fundamental groups of tiles.

X. Bressaud Peano curves and Rauzy fractals and Levitt trees.
Some years ago, Arnoux showed a semi conjugacy between the Rauzy exchange of pieces associated to Tribonacci substitution and a (six) interval(s) exchange transformation. This semi conjugacy yields an immersion of the interval into the Rauzy fractal through a Peano curve. Working with the same interval exchange transformation but a rather different construction, Mac Mullen drew another Peano curve seeming to space fill the Rauzy fractal. With Julien Bernat and Pierre Arnoux, we found an substitutive description of a Peano curve making explicit the link between the two constructions. Moreover, this construction can be generalized in different ways yielding results for other substitutions and a new point of view on the real (Levitt) trees associated to positive free group automorphisms.

A. Fisher unique ergodicity and velocity for adic transformations and solenoids
We show that a two-sided ordered Bratteli diagram naturally defines a solenoid. There are several dualities between the past and future of this nonstationary shift space: the geometrical order on the solenoid leaves is dual to the dynamical order for the adic transformation, which is the return map for the translation flow on the solenoid. Length along leaves can be defined by a scaling function which is dual to Keane's g-measure function. We give a necessary and sufficient condition for the adic transformation to be uniquely ergodic; this is dual to the flow having a unique natural constant velocity.