Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences

摘要

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 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.