# Backward orbits of critical points

## TL;DR

This paper investigates the Galois groups of certain polynomial extensions over number fields, analyzing prime distributions related to $p$-adic attracting points, and applies findings to the Dynamical Mordell-Lang Conjecture.

## Contribution

It introduces a novel analysis of Galois groups of polynomial iterates and connects this to prime distributions and dynamical conjectures in number theory.

## Key findings

- Determines the structure of Galois groups for polynomial iterates.
- Estimates the density of primes with $p$-adic attracting points.
- Applies results to cases of the Dynamical Mordell-Lang Conjecture.

## Abstract

We examine the Galois groups of the extensions $K((f'\circ f^n)^{-1}(0))/K$ where $K$ is a number field for polynomials $f(x)\in K[x]$. We use our understanding of this group to study the proportion of primes for which $f$ has a $\mathfrak p$-adic attracting periodic point for a "typical" $f$ and apply the result to the split case of the Dynamical Mordell-Lang Conjecture.