Consider a matrix $U$ from the unitary group $U_N(\mathbb{C})$ and consider the map $f:U_N(\mathbb{C})\rightarrow U_N(\mathbb{C})$ where $f(U)$ is the matrix of the eigenvectors of $U$.

What is known for the orbits of this map, i.e. $(f^n(U):\;n\in\mathbb N)$ ?

And what can we say about the eigenvalues of each $f^n(U)$ ?

One observation is that if you endow $U_N(\mathbb{C})$ with its Haar measure, then its image by $f$ is still Haar distributed : it kind of preserves disorder.

An other observation is that if you embed $U_{N-1}(\mathbb{C})$ in an obvious way in $U_N(\mathbb{C})$, then $f$ preserves that subgroup.

I guess this has been studied somewhere, but I'm not able to find any reference (but maybe it is just that I don't know a proper name for it).