Discrete dynamics over double commutative-step digraphs

A double-loop digraph, G(N ; s1; s2), has the set of vertices V = ZN and the adjacencies are de0ned by i → i + sk (modN ), k = 1; 2 for any i ∈ V . Double commutative-step digraph generalizes the double-loop. A double commutative-step digraph can be represented by a L-shaped tile, which periodically tessellates the plane. This geometrical approach has been used in several works to optimize some parameters related to double-loops. Given an initial tile L0, we de0ne a discrete iteration L0 → L1 → · · · → Lp → Lp+1 → · · · over L-shapes (equivalently over double commutative-step digraphs). So, we obtain an orbit generated by L0. We classify the set of L-shaped tiles by its behaviour under the above-mentioned discrete dynamics. The study is mainly focussed on the variation of the diameter value of Lp while increasing the value of p. c © 2001 Elsevier Science B.V. All rights reserved.