Substitutions in dynamics, arithmetics and combinatorics. (English) Zbl 1014.11015

Lecture Notes in Mathematics. 1794. Berlin: Springer. xv, 402 p. EUR 57.95/net; sFr. 96.50; £40.50; $ 76.80 (2002).

This collective book, published under the pseudonym N. Pytheas Fogg, based on courses given by the authors in several universities and during several summer schools, addresses the study of substitutions from the triple point of view of dynamics, arithmetics, and combinatorics: from transcendence to partitions, from symbolic (substitutive) dynamical systems to Sturmian sequences, from spectral theory to Diophantine approximations and fractals, from invertible substitutions to polynomial substitutive dynamical systems, from piecewise linear transformations of the unit interval to Cantor sets. The twelve chapters written by ten authors and the appendix written by an eleventh author give an eclectic variety of themes that is both pleasant to read and filled with many results and ideas.
The book ends with a large bibliography of 469 items. Actually the numbers corresponding to references quoted in the chapters after the first are shifted by 1 roughly after [100], but the correctly numbered bibliography is freely accessible at http://link.springer.de/series/lnm.

Reviewer: Jean-Paul Allouche (Orsay)

MSC:

11B85	Automata sequences
11-02	Research exposition (monographs, survey articles) pertaining to number theory
37-02	Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
05-02	Research exposition (monographs, survey articles) pertaining to combinatorics
68Q45	Formal languages and automata
68R15	Combinatorics on words
11A55	Continued fractions
11A63	Radix representation; digital problems
11J70	Continued fractions and generalizations
37B10	Symbolic dynamics
28D99	Measure-theoretic ergodic theory
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
05A17	Combinatorial aspects of partitions of integers

Keywords:

substitutions; transcendence; partitions; dynamical systems; Sturmian sequences; spectral theory; Diophantine approximations; fractals; invertible substitutions; polynomial substitutive dynamical systems; piecewise linear transformations of the unit interval; Cantor sets