Polynomial compositions with large monodromy groups and applications to arithmetic dynamics

For a composition $f=f_1\circ \cdots \circ f_r$ of polynomials $f_i\in \mathbb{Q}[x]$ of degrees $d_i\ge 5$ with alternating or symmetric monodromy group, we show that the monodromy group of $f$ contains the iterated wreath product $A_{d_r}\wr \cdots \wr A_{d_1}$. A similar property holds more generally for polynomials that do not factor through $x^d$ or Chebyshev. We derive consequences to arithmetic dynamics regarding arboreal representations, and forward and backward orbits of such $f$. In particular, given an orbit $(a_n{)}_{n=0}^{\mathrm{\infty }}$ of $f$ as above, we show that for “almost all” $a\in \mathbb{Z}$, the set of primes $p$ for which some $a_n$ is congruent to $a$ mod $p$ is “small”.