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Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly. (English) Zbl 1156.37301
Kolyada, S. (ed.) et al., Algebraic and topological dynamics. Proceedings of the conference, Bonn, Germany, May 1–July 31, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3751-6/pbk). Contemp. Math. 385, 333-364 (2005).
E. M. Coven and G. A. Hedlund [Math. Syst. Theory 7, 138–153 (1973; Zbl 0256.54028)] and M. Morse and G. A. Hedlund [Am. J. Math. 62, 1–42 (1940; Zbl 0022.34003)] exhibited a deep connection between the word combinatorics of Sturmian sequences and the dynamics of irrational circle rotations. G. Rauzy [e.g., in: Sémin. Delange-Pisot-Poitou, 18e Année 1976/77, Théor. des Nombres, Fasc. 1, Exp. 15, 16 p. (1977; Zbl 0369.28015); Acta Arith. 34, 315–328 (1979; Zbl 0414.28018)] showed that the Euclidean algorithm is also naturally connected to these problems, and used this to give an entirely different proof. More than this, Rauzy initiated a program to simultaneously generalize some or all of the vertices of this interesting dynamics – combinatorics – Diophantine approximation triangle. This paper surveys the state of the art in attempts to realize Rauzy’s program. The primary focus is on the connections between Arnoux–Rauzy sequences and rotations of the 2-torus; coding of two-dimensional actions and two-dimensional Sturmian words; and interval exchanges and sequences of low complexity. In addition to a careful account of this rapidly developing field, the paper concludes with an extensive bibliography of some 130 items, making it a valuable resource for other researchers.
For the entire collection see [Zbl 1075.37001].

MSC:
37B10 Symbolic dynamics
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
11A63 Radix representation; digital problems
11J70 Continued fractions and generalizations
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
68R15 Combinatorics on words
05A05 Permutations, words, matrices
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