Universality of free random variables: atoms for non-commutative rational functions

We address the following question: what can one say, for a tuple $(Y_1,\dots,Y_d)$ of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each $Y_i$, about the sizes of the eigenspaces for any non-commutative polynomial $P(Y_1,\dots,Y_d)$ in those operators? We show that for each polynomial $P$ there are unavoidable eigenspaces, which occur in $P(Y_1,\dots,Y_d)$ for any $(Y_1,\dots,Y_d)$ with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms - where it is given by realizations via matrices over the free skew field and via rank calculations - and in analytic terms - where it is given by freely independent random variables with prescribed atoms in their distributions. The fact that the latter situation corresponds to this minimal situation allows to draw many new conclusions about atoms in polynomials of free variables. In particular, we give a complete description of atoms in the free commutator and the free anti-commutator. Furthermore, our results do not only apply to polynomials, but much more general also to non-commutative rational functions.