Global convergence and non existence of periodic points of period 4

It is given a non trivial example of nonempty subset I of C0([0,1]2) such that: whatever F I be, for the pair ([0,1]2,F) the successive approximations method converges globally (i.e. For each P [0,1]2 the sequence (Fn(P))n N converges to a fixed point of F if and only if F has no periodic point of period 4.