energy composed by a number of cycles and fixed points. (b) Cartoon of a cycle of period $T$, for which the cycle is composed of $T$ states. ](CycleDraft.eps "fig:"){width="7cm"}

Master equation {#masterEqn}
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Given a set of initial conditions with a fixed energy $E$, the probability distribution $\varrho^E_{t}(\{ x ,y\})$ evolves following a Perron-Frobenius like-equation $$\varrho^{E}_{t+1} = {\mathcal L}^{E} \varrho^{E}_{t}$$ which, in principle, can be computed after the microscopic rule of evolution (\[q2r\]). Indeed ${\mathcal L}^{E}$ is easy to build: if the state $\{x,y\}_i$ at time $t$ goes to $\{x,y\}_k$, at a time $t+1$, then one sets the $(i,k)$ components to 1, that is ${\mathcal L}^{E}_{ik}=1$. Checking all available elements, $\Omega(E) $, for a given energy we built the huge, $\Omega(E) \times \Omega (E) $, linear operator, ${\mathcal L}^{E}$. This matrix possesses a large number of blocks and zeroes revealing the existence of a large number of cycles in the Q2R model (In some sense, ${\mathcal L}^{E}$ is a kind of adjacency matrix of a graph, the graph being the total number of cycles shown for a given energy).

However, this description is impractical because of the typical magnitude of $\Omega(E)$. Therefore, the full phase space is reduced to a 