Dynamic of cyclic automata over ℤ²

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)modq, ∀αQ. In this work we study the following dynamic F on Q ^Z2. For xQ^Z2 we define F_v(x)=s(x_v) if the state s(x_v) appears in one of the four neighbors of v in K and F_v(x)=x_v otherwise. For xQ^Z2, such that {vZ²: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 lcm 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 k₁,...,k_n of non-negative integers there exists yⁿQ^Z2 such that |{vZ²:yⁿ_v≠0}|=O(k₁ ²+...+k_n²) and the period of the vertex (0,0) is p·lcm{k₁,...,k_n}, for some even integer p.