Discrete rotations and symbolic dynamics

The aim of this paper is to study local configurations issued from discrete rotations. The algorithm of discrete rotations that we consider is the discretized rotation. It simply consists in the composition of a Euclidean rotation with a rounding operation, as studied in [B. Nouvel, E. Rémila, On colorations induced by discrete rotations, in: DGCI, in: LNCS, vol. 2886, 2003, pp. 174–183; B. Nouvel, E. Rémila, Characterization of bijective discretized rotations, in: International Workshop on Combinatorial Images Analysis, 10th International Conference, IWCIA 2004, Auckland, New Zealand, December 1–4, 2004, in: LNCS, vol. 3322, 2004, pp. 248–259; B. Nouvel, E. Rémila, Configurations induced by discrete rotations: Periodicity and quasiperiodicity properties, Discrete Appl. Math. 2–3 (147) (2005) 325–343]. It is possible to encode all the information concerning a discrete rotation as two multidimensional words Cα and C ′ α that we call configurations. In this paper, we introduce two discrete dynamical systems defined by a Z2-action on the two-dimensional torus that allow us to describe the configurations Cα and C ′ α via a suitable symbolic coding; we then deduce various combinatorial properties for both families of configurations, and in particular, results concerning densities of symbol occurrence. c © 2007 Elsevier B.V. All rights reserved.