Let K be the two-dimensional grid. Let q be an integer greater than 1 and let Q={0,...,q-1}. Let s : Q → Q be defined by s(α) = (α + 1) mod q, ∀a ∈ Q. In this work we study the following dynamic F on Q Z2 . For x ∈ Q Z2  we define F v (x) = s(x v ) if the state s(x r ) appears in one of the four neighbors of v in K and F v (x) = x v  otherwise. For x ∈ Q Z2 , such that {v ∈ Z 2 : x v  ¬= 0} is finite we prove that there exists a finite family of cycles such that the period of every vertex of K divides the 1cm of their lengths. This is a generalization of the same result known for finite graphs. Moreover, we show that this upper bound is sharp. We prove that for every n ≥ 1 and every collection k 1 ,...,k n  of non-negative integers there exists y n  ∈ Q Z2  such that |{v ∈ Z 2 : y n v ¬= 0}| = O(k 2 1 + ... + k 2 n) and the period of the vertex (0,0) is p . 1cm{k 1 ,...,k n }, for some even integer p.
['Martín Matamala', 'Eduardo Moreno']
Dynamic of cyclic automata over Z2