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

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

“…We show that, by projecting this stepped surface on the diagonal plane x -~ ~ + z = 0 along the main diagonal direction (1,1,1), and considering the lattice F, projection of 713 on this plane (this lattice is isomorphic to Z2 ), one can code the stepped surface as a two-dimensional sequence U with values in a three-letter alphabet (i.e., a map from Z2 to the set {I, 2, 3}). We then recall [10] how one can recover this sequence as a symbolic dynamics for the Z2 -action by two rotations Ra, and Rb of respective angles a and b on a circle of length a + b + c, and we prove the following result: THEOREM 1. -Let U be the coding of the plane P : ax + by +cz + h = 0, with a, b, c strictly positive.…”

“…We will then explain how we can recover this symbolic sequence as symbolic dynamics of a Z2 -action generated by two rotations on the circle [10]. Our construction can be rephrased in terms of the classical "cut and project" construction (see for instance [39]); see also [41] for a dual approach.…”

Arithmetic Discrete Planes Are Quasicrystals

“…Now one checks that the set of distinguished vertices of P(v, μ, ω) is a lattice (see [BV00]). We thus have found a lattice underlying the arithmetic discrete plane P(v, μ, ||v|| 1 ) even if the coordinates of v are rationally independent, that is, even if the arithmetic discrete plane has no nonzero period vector.…”

“…For a full proof and more details, see [BV00]. In other words, our convention for the choice of a distinguished vertex of a face implies that a face of type i with distinguished vertex x is included in P(v, μ, ω) if and only if x, v + μ ∈ I i .…”

“…We use in this latter case Bezout's lemma. For a proof, see [BV00]. It also uses the fact that I C is described as an intersection of intervals which are preimages of the intervals I i under the action of the translations x → x + v j , for i, j ∈ {1, 2, 3}.…”