matrix does not possess, in general, only real eigenvalues. This subject will be matter of a future research.

Finally, it is important to compare the results for different partition sizes. First, we compute the equilibrium distribution for three partitions sets. More precisely, for a $8\times 8$ system evolving by Q2R at $E=0$ in a periodic orbit of $T=672018$.

Figure \[Method88\] (a) compares the three different coarse-graining partitions (5, 11 and 34 elements of the partition). Despite the evident differences among the coarse and the fine graining partitions, one notices that both partitions possess the same accurate behavior of the equilibrium distribution. Moreover, Figure \[Method88\] (b) compares the second eigenmode ${\bm \chi}^{(2)}$ without any substantial difference among the partitions.

\(a) ![ \[Method88\] (a) Plot of the equilibrium distribution $f_{eq}$ vs $M$ for a $8\times 8$ system with $E=0$ and a cycle of a period $T=672018$ for three different partitions of the magnetization values. The plot shows that all distribution functions lies under the same curve. The inset shows the parabolic behavior in magnetization which after a fit reads $ \log f_{eq}= -M^2/ 116$. (b) Plot of the second eigenmode ${\bm \chi}^{(2)}$ corresponding to the eigenvalue closest to the unit circle. One 