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Coding of a translation of the two-dimensional torus. (English) Zbl 1171.37009
A symbolic coding of the ergodically acting map \(T:x\to x+\alpha\), where \(\alpha\in {\mathbb T}^2\) (\({\mathbb T}^2={\mathbb R}^2 /{\mathbb Z}^2\) is the 2D-torus) is presented. It shares several properties with the Sturmian coding os a one-dimensional translation. Such obtained symbolic dynamical system is metrically isomorphic to \(({\mathbb T}^2,\, T)\). The coding is of quadratic growth complexity and 2-balanced.

MSC:
37B10 Symbolic dynamics
11K60 Diophantine approximation in probabilistic number theory
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