Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

The Geometry Of Fractal Sets

We prove that the dynamical system generated by a primitive unimodular substitution of the Pisot type ond letters satisfying a combinatorial condition which is easy to check, is measurably isomorphic to a domain exchange in R d 1 , and is a nite extension of a translation on the torus T d 1 I n the course of the proof, we introduce some potentially useful notions: the linear maps associated to a substitution and their dual maps, and the -structure for a dynamical system with respect to a pair of partitions

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Pisot substitutions and Rauzy fractals

Young people’s participation in science, technology, engineering and mathematics (STEM) is a matter of international concern. Studies and careers that require physical sciences and advanced mathematics are most affected by the problem and women in particular are under-represented in many STEM fields. This article views international research about young people’s relationships to, and participation in, STEM subjects and careers through the lens of an expectancy value model of achievement-related choices. In addition it draws on sociological theories of late-modernity and identity, which situate decision-making in a cultural context. The article examines how these frameworks are useful in explaining the decisions of young people – and young women in particular – about participating in STEM and proposes possible strategies for removing barriers to participation.

https://www.naturfagsenteret.no/binfil/download2.php?tid=1657187

Participation in science and technology: young people’s achievement‐related choices in late‐modern societies

This paper surveys different constructions and properties of some multiple tilings (that is, finite-to-one coverings) of the space that can be associated with beta-numeration and substitutions. It is indeed possible, generalizing Rauzy’s and Thurston’s constructions, to associate in a natural way either with a Pisot number β (of degree d) or with a Pisot substitution σ (on d letters) some compact basic tiles that are the closure of their interior, that have non-zero measure and a fractal boundary; they are attractors of some graph- directed Iterated Function System. We know that some translates of these prototiles under a Delone set Γ (provided by β or σ) cover Rd−1; it is conjectured that this multiple tiling is indeed a tiling (which might be either periodic or self-replicating according to the translation set Γ). This conjecture is known as the Pisot conjecture and can also be reformulated in spectral terms: the associated dynamical systems have pure discrete spectrum. We detail here the known constructions for these tilings, their main properties, some applications, and focus on some equivalent formulations of the Pisot conjecture, in the theory of quasicrystals for instance. We state in particular for Pisot substitutions a finiteness property analogous to the well-known (F) property in beta-numeration, which is a sufficient condition to get a tiling.

https://hal-lirmm.ccsd.cnrs.fr/lirmm-00105299/document

Tilings associated with beta-numeration and substitutions.

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 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.

Higher dimensional extensions of substitutions and their dual maps

Nous definissons des substitutions bi-dimensionnelles ; ces substitutions engendrent des suites doubles reliees a des approximations discretes de plans irrationnels. Elles sont obtenues au moyen de l'algorithme classique de Jacobi Perron, en definissant l'induction d'une action de Z 2 par rotations sur le cercle. On donne ainsi une interpretation geometrique nouvelle de l'algorithme de Jacobi-Perron, comme application operant sur l'espace des parametres des actions de Z 2 par rotations.

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Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions

Two-dimensional iterated morphisms and discrete planes

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 of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.

/pdf/tilings-quasicrystals-discrete-planes-generalized-2d2fbnem7y.pdf

Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

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 and the invariant measure for each of the additive and multiplicative versions of this algorithm.

Pierre Arnoux

Papers

Higher dimensional extensions of substitutions and their dual maps

Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions

Two-dimensional iterated morphisms and discrete planes

Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

Fractions continues sur les surfaces de Veech