is a hypercube in dimension 8, however the dynamics is too simple, it does contain cycles of period 4 at most. The phase space of a $4\times 4$ system is the largest possible one that can be studied exactly case by case. In this case the system possesses $2^{2\times 16}=2^{32}$ states and it contains a rich variety of cycles (see Table \[Example4x4\]). This case will be studied deeply as a good benchmark for conjectures in larger dimensional systems.

As an example, from this case, it is observed that the total number of cycles, $n(T,E)$, of period $T$ and energy $E$, would be bounded by [@cycles] $$n(T,E) < \frac{1}{T} 2^{2N} e^{-\alpha |E|} \sim e^{2 N \log 2 - \alpha |E|}.$$ From the data one has that for $3\times 3$ and $4\times 4$, $\alpha \approx 0.6$, but this value varies as the lattice size increases. Here one notices a dramatic difference among the cases $|E|$ greater or smaller than $E_c=\frac{2}{\alpha} \log 2$, if $|E|$ is greater than $E_c$ the probability to see a long period is exponentially small, but on the contrary, for $|E|<E_c$ this probability reaches the unity. Higher lattice sizes confirm this scenario but modify slightly the value of $\alpha$. 