Substitutions in Dynamics, Arithmetics and Combinatorics

N. Pytheas Fogg, Valéré Berthé, Sébastien Ferenczi, Christian Mauduit, Anne Siegel (eds.)3540441417, 9783540441410

A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the ‘simplest’ possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure.
The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a state-of-the-art survey of the more difficult and unsolved problems.

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Table of contents :
Basic notions on substitutions….Pages 1-32
Substitutions, arithmetic and finite automata: an introduction….Pages 35-52
Automatic sequences and transcendence….Pages 53-80
Substitutions and partitions of the set of positive integers….Pages 81-98
Substitutions and symbolic dynamical systems….Pages 101-142
Sturmian Sequences….Pages 143-198
Spectral theory and geometric representation of substitutions….Pages 199-252
Diophantine approximations, substitutions, and fractals….Pages 253-292
Infinite words generated by invertible substitutions….Pages 295-320
Polynomial dynamical systems associated with substitutions….Pages 321-342
Piecewise linear transformations of the unit interval and Cantor sets….Pages 343-361
Some open problems….Pages 363-374
A. Undecomposable matrices in dimension 3 (by J. Rivat)….Pages 375-376