Pierre Arnoux

Institut de Mathématiques de Luminy | IML · Mathématiques

This book presents a panorama of recent developments in the theory of tilings and related dynamical systems. It contains an expanded version of courses given in 2017 at the research school associated with the Jean-Morlet chair program. Tilings have been designed, used and studied for centuries in various contexts. This field grew significantly aft...

The Arnoux-Rauzy systems are defined in \cite{ar}, both as symbolic systems on three letters and exchanges of six intervals on the circle. In connection with a conjecture of S.P. Novikov, we investigate the dynamical properties of the interval exchanges, and precise their relation with the symbolic systems, which was known only to be a semi-conjuga...

We give a heuristic method to solve explicitly for an absolutely continuous invariant measure for a piecewise differentiable, expanding map of a compact subset $I$ of Euclidean space $R^d$. The method consists of constructing a skew product family of maps on $I\times R^d$, which has an attractor. Lebesgue measure is invariant for the skew product f...

We give an elementary proof of a property discovered by Xavier Grandsart: let W be a circular binary word; then the differences in the number of occurrences /W/0011 - /W/1100, /W/1101 - /W/1011, /W/1010 - /W/0101 and /W/0100 - /W/0010 are equal; this property is easily generalized using the De Bruijn graph.

We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira (1993). We also apply the same method on the unsorted version of Brun algorithm, and illustrate some experimentations on the domain of the natural extens...

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conju...

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and t...

Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the stepped line associated with an element of the symbolic system on the contracting space of the matrix. We show th...

We define the Rauzy gasket as a subset of the standard two-dimensional simplex associated with letter frequencies of ternary epi-Sturmian words. We prove that the Rauzy gasket is homeomorphic to the usual Sierpiński gasket (by a two-dimensional generalization of the Minkowski question mark function) and to the Apollonian gasket (by a map which is s...

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and t...

We define a geometrical continued fraction algorithm in the setting of regular polygons with an even number of sides., The definition of the algorithm uses linear transformations generating a group conjugated to an index 2 subgroup of a Hecke group. We give Markov conditions allowing the iteration of the algorithm. We compute the natural extension...

There has been much recent work on the geometric representation of Pisot substitu-tions and the explicit construction of Markov partitions for Pisot toral automorphims. We give a construction that extends this to the general hyperbolic case. For the sake of simplicity, we consider a simple example of an automorphism of the free group on 4 generator...

We consider a substitution associated with the Arnoux–Yoccoz interval exchange transformation (IET) related to the tribonacci substitution.We construct the so-called stepped lines associated with the fixed points of the substitution in the abelianization (symbolic) space.We analyze various projections of the stepped line, recovering the Rauzy fract...

The role of mental representations in mathematics and computer sciences (for teaching or searching) is often downplayed or even completely ignored. Using an ongoing work on the subject, we argue for a more systematic study and use of mental representations, to get an intuition of mathematical concepts, and also to understand and build proofs. We gi...

Veech’s original examples of translation surfaces Vq enjoying what McMullen has dubbed “optimal dynamics ” arise from appropriately gluing sides of two copies of the regular q-gon, with q ≥ 3. We show that every Vq whose trace field is of degree greater than 2 has non-periodic directions of vanishing SAF-invariant. (Calta-Smillie have shown that un...

In this article, we start by considering the relevant parts of the French educational system and the data relating to science at the end of secondary school and the early years of university. We show an increase, till 1995, and then a decline in the study of mathematics at baccalaureate and university level. However, our main conclusions relate to...

In this paper, we give a necessary condition for an infinite word defined by a non-degenerate interval exchange on three intervals (3iet word) to be invariant by a substitution: a natural parameter associated to this word must be a Sturm number. We deduce some algebraic consequences from this condition concerning the incidence matrix of the associa...

A substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words is proved to be well-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated with any usu...

We consider the problem of generation of discrete planes using generalized substitutions. We give sufficient conditions to be sure to generate all of a discrete plane by a sequence of substitutions; these conditions, however, are not easy to check, even on simple examples. One can build approximations of discrete planes in several ways, namely as s...

In this paper, we will first formulate and prove some equivalent sufficient conditions to obtain the tiling property for a Pisot unimodular substitution. We will then apply these condition to the more general framework of adic systems, to extend to this more general (and non algebraic) case results already known for the substitutive case.

In this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., 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...

A substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words introduced in [AI01,ABS04] is proved to be sell-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can...

We introduce the notion of an Anosov family, a generalization of an Anosov map of a manifold. This is a sequence of diffeomorphisms along compact Riemannian manifolds such that the tangent bundles split into expanding and contracting subspaces. We develop the general theory, studying sequences of maps up to a notion of isomorphism and with respect...

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 r...

The paper concerns the C -closing lemma (r 2) for vector fields with finitely many singularities on an orientable hyperbolic compact surface M which might have nonempty boundary. Let M ae Delta be an universal cover of M , where Delta is the hyperbolic plane, and let S1 be the circle at infinity of Delta. Given a nontrivially recurrent semitrajecto...

Given a substitution σ ond letters, we define itsk-dimensional extension,E k (σ), for 0≤k≤d. Thek-dimensional extension acts on the set ofk-dimensional faces of unit cubes inR 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 bou...

We introduce two-dimensional substitutions generating two-dimensional sequences re-lated to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a Z,-action by rotations on the circle. This gives a new geometric interpre...

this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms. The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show...

We prove that the dynamical system generated by a primitive unimodular substitution of the Pisot type on d letters satisfying a combinatorial condition which is easy to check, is measurably isomorphic to a domain `exchange in Rd-1, and is a finite extension of a translation on the torus Td-1. I, the course of the proof, we introduce some potentiall...

The aim of this paper is to give an overview of recent results a bout tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms. The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution,...

In the field of dynamical systems, it is customary to oppose ordered dynamical systems, and chaotic dynamical systems. This opposition can be expressed in several different ways: systems of entropy zero versus systems of strictly positive entropy, systems with low (polynomial) complexity versus systems with exponential complexity, systems without o...

International audience The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration...

We define the scenery flow of the torus. The flow space is the union of all flat 2-dimensional tori of area 1 with a marked direction (or equivalently, on the union of all tori with a quadratic differential of norm 1). This is a 5-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic diff...

R´ ESUM´ E.—N ous demontrons une conjecture de Gerard Rauzy relativeal a structure des trajectoires de billard dans un cube. A chaque trajectoire on associe la suitea valeurs dans {1,2,3} obtenue en codant par un 1 (resp. 2, 3 )c haque rebond sur une paroi frontale (resp. laterale, horizontale). Nous montrons que si la direction initiale est totale...

Nous donnons une preuve élémentaire et explicite du fait que le flot géodésique sur la surface modulaire (quotient du plan hyperbolique par l'action de SL(2, Z)) peut être codé en utilisant les fractions continues. Abstract. We give an elementary and explicit proof of the coding of the géodésie flow on the modular surface by continued fractions. 0.

Résumé: on définit la complexité d’une suite à valeurs dans un ensemble fini et on montre comment, dans le cas d’une suite définie par le codage d’un système dynamique, on peut calculer explicitement la complexité, en donnant quelques exemples. Abstract : we define the complexity of a sequence taking values in a finite set and we show how, in the...

We give an interpretation of the Fibonacci multiplication defined by D.E. Knuth

We study the relations between the de Rham cohomology class of a closed differential 1-form ω with Morse singularities on a manifoldM of dimensionn≧3, and the ergodic properties of the foliationF ω it defines. We show by examples that, if the fundamental group is “large” enough, very different behaviours can occur in the same class. In contrast, if...

Every aperiodic measure-preserving transformation can be obtained by a cutting and stacking construction. It follows that all such transformations are infinite interval exchanges. This in turn is used to represent any ergodic measure-preserving flow as aC ∞-flow on an open 2-manifold. Several additional applications of the basic theorems are also g...

The paper concerns the C r -closing lemma (r 2) for vector elds with nitely many singularities on an orientable hyperbolic compact surface M which might have nonempty boundary. Let M be an universal cover of M, where is the hyperbolic plane, and let S 1 be the circle at innnity of . Given a nontrivially recurrent semitrajectory l of a ow on M, the...

We give a construction that extends to the general hyperbolic case recent work on the geometric representation of Pisot substitutions and the explicit construction of Markov partitions for Pisot toral automorphisms . For the sake of simplicity, we consider a simple example of an automorphism of the free group on 4 generators, whose associated matri...

Unimodular Substitutions on 2 letters. Conjecture: the dynamical system associated with a primitive substitution on 2 letters, with matrix in SL(2, Z), is measurably isomorphic to a circle rotation. There is a very convenient criterium, due to B.Host: Definition: the substitution σ has strong coincidence if there exists n and k such that σ n (0) an...