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Arnoux, Pierre; Berthé, Valérie; Ito, Shunji

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

Ann. Inst. Fourier 52, No. 2, 305-349 (2002).
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.
Reviewer: Takao Komatsu (Tsu, Mie)

Cited in 1 Review
Cited in 25 Documents

MSC:

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

Keywords:

substitutions; generalized continued fractions; discrete planes; tilings; Jacobi-Perron algorithm; induction; \(\mathbb Z^2\)-actions; two-dimensional sequences
PDFBibTeX XMLCite
\textit{P. Arnoux} et al., Ann. Inst. Fourier 52, No. 2, 305--349 (2002; Zbl 1017.11006)
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