**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

AMS 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.

#### Readership

Graduate students and research mathematicians interested in Pisot substitutions.