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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).

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Your map is not well-defined. Even if $U$ has distinct eigenvalues we are still free to multiply the eigenvectors by complex numbers of absolute value one, or to permute them. There is even more freedom if eigenvalues are repeated. Different choices of $f(U)$ will give completely unrelated answers for $f^2(U)$. –  Neil Strickland Jun 5 '13 at 21:00

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