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Double sequences of low complexity. (Suites doubles de basse complexité.) (French) Zbl 1018.37010
M. Morse and G. A. Hedlund proved that a sequence containing at most $$n$$ blocks of length $$n$$ for some $$n$$ must be periodic from some point on [Am. J. Math. 60, 815–866 (1938; Zbl 0019.33502)]. They also introduced Sturmian sequences [see ibid. 62, 1–42 (1940; Zbl 0022.34003)]: these are the sequences with exactly $$n+1$$ blocks of length $$n$$ for each $$n\geq 1$$. The corresponding thresholds and the “right” notion of Sturmian sequences in dimension $$\geq 2$$ are not really known yet. In the paper under review the authors study the 2D sequences that have $$mn+n$$ rectangular blocks of size $$(m,n)$$ and that are uniformly recurrent. They show that these sequences code the $$\mathbb Z^2$$-action defined by two irrational rotations on $$\mathbb R/\mathbb Z$$. Sturmian sequences occur in the proof. In passing the authors study the 2D sequences that contain $$m+n$$ blocks of size $$(m,n)$$.
Please note that Reference [4] has appeared in Discrete Math. with a slightly modified title (see Zbl 0970.68124), that Reference [10] has appeared (see Zbl 1001.68093), that Reference [15] has appeared (see Zbl 1005.68118), and that Reference [17] has also appeared (see Zbl 0983.68062).

##### MSC:
 37B10 Symbolic dynamics 11B83 Special sequences and polynomials 68R15 Combinatorics on words
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##### References:
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