Topological properties of Rauzy fractals.

*(English)*Zbl 1229.28021The present monograph deals with topological and geometric properties of substitutions. Substitutions are combinatorial objects which produce sequences by iteration (most fractals are obtained by an iterating process). In particular, the authors study some important properties of Rauzy fractals associated with substitutions. Rauzy fractals play a major role in several branches of mathematics (e.g., dynamical systems, combinatorics, number theory) and are useful in some parts of computer science.

This book contains a systematic study of the topological structure and properties of Rauzy fractals based on an ample bibliography. After a thorough survey on basic results for these fractals, know results on the topology of Rauzy fractals are collected and a variety of new ones is proved. In particular, the authors investigate the tiling properties, connectivity, homeomorphy to a closed disk as well as the fundamental group of Rauzy fractals. Graphical illustrations and many examples contribute to a better understanding of the text. The monograph contains a perspective on further research related to this topics, is very well written and the subject is up-to-date.

This book contains a systematic study of the topological structure and properties of Rauzy fractals based on an ample bibliography. After a thorough survey on basic results for these fractals, know results on the topology of Rauzy fractals are collected and a variety of new ones is proved. In particular, the authors investigate the tiling properties, connectivity, homeomorphy to a closed disk as well as the fundamental group of Rauzy fractals. Graphical illustrations and many examples contribute to a better understanding of the text. The monograph contains a perspective on further research related to this topics, is very well written and the subject is up-to-date.

Reviewer: Nicolae-Adrian Secelean (Sibiu)

##### MSC:

28A80 | Fractals |

11A63 | Radix representation; digital problems |

54F65 | Topological characterizations of particular spaces |