Topological properties of Rauzy fractals

Topological properties of Rauzy fractals
[Propriétés topologiques des fractals de Rauzy]

Siegel, Anne ; Thuswaldner, Jörg M.

Mémoires de la Société Mathématique de France, no. 118 (2009) , 140 p.

Résumé
Abstract

Les fractals de Rauzy apparaissent dans diverses branches des mathématiques telles que la théorie des nombres, les systèmes dynamiques, la combinatoire et la théorie des quasi-cristaux. De nombreuses questions font alors intervenir la structure topologique des fractals. Cette monographie propose une étude systématique des propriétés topologiques des fractals de Rauzy. Les premiers chapitres de ce document rappellent les enjeux mathématiques relatifs aux fractals de Rauzy ainsi que les principaux résultats connus à leur sujet. Sont ensuite discutés des propriétés de pavages, de connexité, d’homéomorphisme à un disque, ainsi que le groupe fondamental de ces ensembles. Les méthodes s’appuient sur des résultats en topologie du plan et sur la construction de graphes pour décrire la structure des pavages associés aux fractals. De nombreux exemples caractéristiques sont présentés. Un chapitre final discute des principales perspectives de recherches liées à cette thématique.

Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, in particular, we study properties of the Rauzy fractals associated to substitutions. To be more precise, let $σ$ be a substitution over the finite alphabet $𝒜$ . We assume that the incidence matrix of $σ$ is primitive and that its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to $σ$ a set $𝒯$ which is called central tile or Rauzy fractal of $σ$ . Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in $n = | 𝒜 |$ subtiles $𝒯 (1), ..., 𝒯 (n)$ . The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary. Pisot substitutions and central tiles are of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix representations. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and the geometry of the underlying central tile. After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of $𝒯$ and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether $𝒯$ and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a closed disk and derive properties of their fundamental group. The basic tools for our criteria are several classes of graphs built from the description of the tiles $𝒯 (i)$ ( $1 \leq i \leq n$ ) as the solution of a graph directed iterated function system and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. For instance, they can be used to construct the boundaries $\partial 𝒯$ as well as $\partial 𝒯 (i)$ ( $1 \leq i \leq n$ ) and all points where two, three or four different tiles of the induced tilings meet. When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for “everyday’s life” when dealing with topological and geometric properties of substitutions. Many examples are given throughout the text in order to illustrate our results. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

MR Zbl | 2 citations dans Numdam

DOI : 10.24033/msmf.430

Classification : 28A80, 11A63, 54F65
Keywords: Rauzy fractal, tiling, beta-numeration, connectivity, homeomorphy to a disk, fundamental group
Mot clés : Fractal de Rauzy, Pavage, beta-numération, connexité, homéomorphisme à un disque, groupe fondamental

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Siegel, Anne; Thuswaldner, Jörg M. Topological properties of Rauzy fractals. Mémoires de la Société Mathématique de France, Série 2, no. 118 (2009), 140 p. doi : 10.24033/msmf.430. http://numdam.org/item/MSMF_2009_2_118__1_0/

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