**Memoires de la Societe Mathematique de France**

Volume: 118;
2009;
144 pp;
Softcover

MSC: Primary 28; 11; 54;
**Print ISBN: 978-2-85629-290-7
Product Code: SMFMEM/118**

List Price: $42.00

Individual Member Price: $33.60

# Topological Properties of Rauzy Fractals

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*Anne Siegel; Jörg M. Thuswaldner*

A publication of the Société Mathématique de France

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 this monograph the authors deal with topological and
geometric properties of substitutions; in particular, they study
properties of the *Rauzy fractals* associated to
substitutions.

To be more precise, let \(\sigma\) be a substitution over
the finite alphabet \(\mathcal {A}\). The authors assume that
the incidence matrix of \(\sigma\) 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 \(\sigma\) a set
\(\mathcal{T}\) which is called *central tile* or
*Rauzy fractal* of \(\sigma\). Such a central tile is a
compact set that is the closure of its interior and decomposes in a
natural way in \(n=|\mathcal {A}|\) subtiles \(\mathcal{T}
(1),\ldots ,\mathcal{T} (n)\). 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 such as 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 raised 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 authors investigate a variety of topological properties of \(\mathcal{T}\) and its subtiles. Their approach is an algorithmic one. In particular, they address the question whether \(\mathcal{T}\) 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 the authors' criteria are several classes of graphs built from the description of the tiles \(\mathcal{T} (i)\) (\(1\le i\le n\)) 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 \(\partial \mathcal{T}\) as well as \(\partial \mathcal{T} (i)\) (\(1\le i\le n\)) 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 this monograph aims to provide tools for "everyday life" when dealing with topological and geometric properties of substitutions.

Throughout the text, the authors give many examples to illustrate their results and also offer suggestions for further research.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Table of Contents

# Table of Contents

## Topological Properties of Rauzy Fractals

#### Readership

Graduate students and research mathematicians interested in Pisot substitutions.