This behavior is consistent with the numerical simulations of Ref. [@hans2].

Scope of the paper {#scope}
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Though the Q2R model is quite simple its dynamics is usually very rich as it has been documented extensively in literature. Moreover, this conservative and reversible system appears to behave as a typical macroscopic system, as the number of degrees of freedom increases, showing, in particular, a typical irreversible behavior, sensibility of the initial conditions, a kind of mixing, etc. It is believed that this Q2R is a good representation of an Ising model in thermodynamical equilibrium.

The phase space of the Q2R system of $N$ sites possesses $2^{2 N} $ states, which is partitioned in different sub-spaces of constant energies, which itself are partitioned in a large amount of smaller subspaces of periodic orbits or fixed points. Notice that, because the system is conservative, there are neither attractive nor repulsive attractors, all attractors are fixed points or cycles.

This feature of the phase space is schematized in Fig. \[CycleDraft\] (a), where the constant energy subspace shares in principle many cycles and fixed points. An arbitrary initial condition of energy $E$ falls into one of these cycles, and it runs until it comes 