conservative and explicit reversible cellular automata. In particular, we consider the Q2R model, introduced by Vichniac in the mid 80s [@vichniac], which is a cellular automata that runs on a two dimensional grid of finite size, and is reversible in a physical sense, that is, the automata rule is not only invertible, but the backward rule reads exactly the same as the forward one. Moreover, it was shown by Pomeau [@pomeau84] that the Q2R automata possesses a conserved energy-like quantity.

The main reason to apply the coarse-graining approach to a cellular automata instead to a coupled system of ordinary differential equations, is because a cellular automata is a discrete model with boolean entities as microscopic variables, thus, the system is numerically reversible and conservative. In consequence, Q2R seems to be a good benchmark to test the principles of statistical physics. But, the phase space is finite, hence the dynamical system only possesses fixed points and periodic orbits, therefore it cannot be ergodic, at least in the usual sense of continuos dynamics. However, for large enough systems, the phase space becomes huge, and the periodic orbits may be, as we show, exponentially long, so in practice, of infinite period. Further, if 