Pierre Arnoux

Higher dimensional extensions of substitutions and their dual maps

Given a substitution σ ond letters, we define itsk-dimensional extension,Ek (σ), for 0≤k≤d. Thek-dimensional extension acts on the set ofk-dimensional faces of unit cubes inRd 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 boundary. We say that a substitution is unimodular (resp. hyperbolic) if the matrix associated to the substitution by abelianization is unimodular (resp. hyperbolic). In the case where the substitution is unimodular, we also define dual substitutions which satisfy a similar coboundary condition. We use these constructions to build self-similar sets on the expanding and contracting space for an hyperbolic substitution.

Two-dimensional iterated morphisms and discrete planes

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.

Functional stepped surfaces, flips, and generalized substitutions

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 usual unimodular substitution. The aim of this paper is to extend the domain of definition of such multidimensional substitutions to functional stepped surfaces. One central tool for this extension is the notion of flips acting on tilings by lozenges of the plane.

Geometrical Models for Substitutions

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 fractal, a Peano curve related to work in [Arnoux 88], another Peano curve related to the work of [McMullen 09] and [Lowenstein et al. 07], and also the interval exchange transformation itself.

The Rauzy Gasket

We define the Rauzy gasket as a subset of the standard two-dimensional simplex associated with letter frequencies of ternary episturmian words. We prove that the Rauzy gasket is homeomorphic to the usual Sierpinski gasket (by a two-dimensional generalization of the Minkowski ? function) and to the Apollonian gasket (by a map which is smooth on the boundary of the simplex). We prove that it is also homothetic to the invariant set of the fully subtractive algorithm, hence of measure 0.

Using mental imagery processes for teaching and research in mathematics and computer science

The role of mental representations in mathematics and computer science (for teaching or research) 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 give two detailed examples.

The rise and fall of mathematical enrolments in the French educational system: a case study

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 a decline in the study of all sciences, rather than just mathematics. The reason that we have to consider all sciences is because of the difficulty in getting accurate figures for mathematical enrolments at the university level. We do, however, have accurate figures for high school level mathematics enrolments. We consider the reasons for this downturn in enrolments, which appear to be largely political and social. It is also noted that there have been declines in the numbers of students in areas other than science. Finally, we show that for French tertiary students, well-paid secure jobs are at least as important as subject choice. This is dramatically illustrated by the numbers of applicants applying for competitive positions as secondary teachers.

Random product of substitutions with the same incidence matrix

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 that this Rauzy fractal depends continuously on the sequence of substitutions, and investigate some of its properties; in some cases, this construction gives a geometric model for the symbolic dynamical system.

Note on occurrences of factors in circular words

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.