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Topological Properties of Rauzy Fractals
Anne Siegel, Université de Rennes, France, and Jörg M. Thuswaldner, University of Leoben, Austria
A publication of the Société Mathématique de France.
cover
Mémoires de la Société Mathématique de France
2009; 144 pp; softcover
Number: 118
ISBN-10: 2-85629-290-9
ISBN-13: 978-2-85629-290-7
List Price: US$42
Member Price: US$33.60
Order Code: SMFMEM/118
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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.

Table of Contents

  • Introduction
  • Substitutions, central tiles and beta-numeration
  • Multiple tilings induced by the central tile and its subtiles
  • Statement of the main results: Topological properties of central tiles
  • Graphs that contain topological information on the central tile
  • Exact statements and proofs of the main results
  • Technical proofs and definitions
  • Perspectives
  • Bibliography
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