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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.

11B85Automata sequences
11-02Research monographs (number theory)
37-02Research exposition (Dynamical systems and ergodic theory)
05-02Research monographs (combinatorics)
68Q45Formal languages and automata
68R15Combinatorics on words
11A55Continued fractions (number-theoretic results)
11A63Radix representation; digital problems
11J70Continued fractions and generalizations
37B10Symbolic dynamics
28D99Measure-theoretic ergodic theory
37A45Relations of ergodic theory with number theory and harmonic analysis
05A17Partitions of integers (combinatorics)