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Discrete planes, \({\mathbb Z}^2\)-actions, Jacobi-Perron algorithm and substitutions. (English) Zbl 1017.11006

This paper discusses an explicit method to build a discrete approximation of an irrational plane in \(\mathbb R^3\). This approximation can be described by a two-dimensional sequence, which is directly related to symbolic dynamics for a \(\mathbb Z^2\)-action by rotations on the unit circle. This sequence can be generated by applying the Jacobi-Perron algorithm to the coordinates of the unit vector orthogonal to the given plane. This paper attempts to generalize to higher dimensions well-known results for the usual continued fractions.

MSC:

11A55 Continued fractions
11J70 Continued fractions and generalizations
40A15 Convergence and divergence of continued fractions
68R15 Combinatorics on words
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