  A polynomial with integer coefficients yields a family of dynamical systems
indexed by primes as follows: for any prime $p$, reduce its coefficients mod
$p$ and consider its action on the field $\mathbb{F}_p$. We say a subset of
$\mathbb{Z}[x]$ is dynamically distinguishable mod $p$ if the associated mod
$p$ dynamical systems are pairwise non-isomorphic. For any
$k,M\in\mathbb{Z}_{>1}$, we prove that there are infinitely many sets of
integers $\mathcal{M}$ of size $M$ such that $\left\{ x^k+m\mid
m\in\mathcal{M}\right\}$ is dynamically distinguishable mod $p$ for most $p$
(in the sense of natural density). Our proof uses the Galois theory of
dynatomic polynomials largely developed by Morton, who proved that the Galois
groups of these polynomials are often isomorphic to a particular family of
wreath products. In the course of proving our result, we generalize Morton's
work and compute statistics of these wreath products.
