On roots of unity in orbits of rational functions

We consider a large class of univariate rational functions over a number field $\mathbb{K}$, including all polynomials over $\mathbb{K}$, and give a precise description of the exceptional set of such functions $h$ for which there are infinitely many initial points in the cyclotomic closure $\mathbb{K}^c$ for which the orbit under iterations of $h$ contains a root of unity. Our results are similar to previous results of Dvornicich and Zannier describing all polynomials having infinitely many preperiodic points in $\mathbb{K}^c$. We also pose several open questions.