the initial state is random, the temporal behavior may be quite random and it possesses many aspects of chaotic systems, as sensibility to initial conditions, mixing, etc. For any purpose, the observation of a short periodic orbit is really improbable for large enough systems with random initial conditions. In general, there is a huge number of initial conditions that are “almost” ergodic. Numerical studies show that the premises of statistical physics are valid, in particular the observables may be computed using the standard methods of statistical physics.

The study of the dynamics and properties of the Q2R model has a long history. Soon after the seminal works of Vichniac [@vichniac] and Pomeau [@pomeau84], Herrmann[@hans] implemented the Q2R algorithm to study the two space dimensional Ising model in the frame of the micro-canonical ensemble. He studied the global magnetization, obtaining an excellent representation for the magnetization as a function of the initial conserved energy, displaying a coherent picture for the phase transition of the Ising model. Later, Herrmann, Carmesin and Stauffer [@hans2] studied numerically the probability to reach an “infinitely” long period for some energies. Moreover, if the energy is large enough this probability tends to unity [@hans2]. Next, Takesue [@takesue] 