Discrete dynamics over double commutative-step digraphs Original Research Article

A double-loop digraph, G(N;s1,s2), has the set of vertices V=ZN and the adjacencies are defined by i→i+sk (mod N), 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 define a discrete iterationL0→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.