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Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in\mathbb{P}^1(K)$, we study the iterated extension $K(\phi^{-\infty}(b))$generated by the union over $n$ of all solutions to $\phi^n(x) = b$. We askwhen a finite extension of $K(\phi^{-\infty}(b))$ can contain all $m$-powerroots of unity for some $m \geq 2$, and prove that several families of rationalfunctions do so. A motivating application is to understand the higherramification filtration when $K$ is a finite extension of $\Q_p$ and $p$divides the degree of $\phi$, especially when $\phi$ is post-critically finite(PCF). We show that all higher ramification groups are infinite for newfamilies of iterated extensions, for example those given by bicritical rationalfunctions with periodic critical points. We also give new examples of iteratedextensions with subextensions satisfying an even strongerramification-theoretic condition called arithmetic profiniteness. We conjecturethat every iterated extension arising from a PCF map should have a subextensionwith this stronger property, which would give a dynamical analogue of Sen'stheorem for PCF maps.

Roots of unity and higher ramification in iterated extensions