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Substitution dynamical systems: Algebraic characterization of eigenvalues. (English) Zbl 0866.11023
The authors study the eigenvalues of dynamical systems arising from primitive morphisms of the free monoid (substitutions). This theme has been addressed in several papers, in particular by Dekking, Host, Livshits, Vershik, Solomyak. For a good exposition of the works before 1987 see the book by M. Quéffelec [Substitution dynamical systems – spectral analysis, Lect. Notes Math. 1294, Springer Verlag (1987; Zbl 0642.28013)]. In the paper under review the authors give an algebraic characterization of the eigenvalues (too technical to be described here). They use this result to obtain a necessary and sufficient condition for a primitive substitution to be weakly mixing.
Minor remarks: The word “periodical” is systematically used instead of “periodic”; one might note that in the examples of Section 4 the number \(\sqrt{7\pm 2\sqrt{10}}\) can be safely replaced by \(\sqrt{5}\pm \sqrt{2}\); finally, Reference [MOS] appeared, and with a slightly modified title [B. Mossé, Reconnaissabilité des substitutions et complexité des suites automatiques, Bull. Soc. Math. Fr. 124, 329-346 (1996; Zbl 0855.68072)].

MSC:
11B85 Automata sequences
37E99 Low-dimensional dynamical systems
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