Statistics of Young diagrams of cycles of dynamical systems for finite tori automorphisms

A permutation of a set of $N$ elements is decomposing this set into $y$ cycles of lengths $x_s$, defining a partition $N=x_1+\dots+x_y$. The length $X_1$, the height y and the fullness $\lambda=N/xy$ of the Young diagram $x_1\geq x_2\ge\dots\ge x_y$ behave for the large random permutation like $x\sim an$, $y\sim b\ln N$, $\lambda\sim c/\ln N$.
The finite 2-torus $M$ is the product $\mathbb Z_m\times\mathbb Z_m$, and its Fibonacci automorphism sends $(u,v)$ to $(2u+v,u+v)$ (mod $m$). This permutation of $N=m^2$ points of the finite torus $M$ defines a peculiar Young diagram, whose behavior (for large $m$) is very different from that of a random permutation of $N$ points.