Tilings and rotations on the torus: A two-dimensional generalization of Sturmian sequences. (English) Zbl 0970.68124

Berthé, Valérie; Vuillon, Laurent

Discrete Math. 223, No. 1-3, 27-53 (2000).

Summary: We study a two-dimensional generalization of Sturmian sequences corresponding to an approximation of a plane: these sequences are defined on a three-letter alphabet and code a two-dimensional tiling obtained by projecting a discrete plane. We show that these sequences code a \(\mathbb{Z}^2\)-action generated by two rotations on the unit circle. We first deduce a new way of computing the rectangle complexity function. Then we provide an upper bound on the number of frequencies of rectangular factors of given size.

MSC:

68R15

Combinatorics on words

Keywords:

Sturmian sequences