  Given a finite endomorphism $\varphi$ of a variety $X$ defined over the field
of fractions $K$ of a Dedekind domain, we study the extension
$K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1} K(\varphi^{-n}(\alpha))$
generated by the preimages of $\alpha$ under all iterates of $\varphi$. In
particular when $\varphi$ is post-critically finite, i.e., there exists a
non-empty, Zariski-open $W \subseteq X$ such that $\varphi^{-1}(W) \subseteq W$
and $\varphi : W \to X$ is \'etale, we prove that
$K(\varphi^{-\infty}(\alpha))$ is ramified over only finitely many primes of
$K$. This provides a large supply of infinite extensions with restricted
ramification, and generalizes results of Aitken-Hajir-Maire in the case $X =
\mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = \mathbb{P}^1$.
Moreover, we conjecture that this finite ramification condition characterizes
post-critically finite morphisms, and we give an entirely new result showing
this for $X = \mathbb{P}^1$. The proof relies on Faltings' theorem and a local
argument.
