The critical orbit structure of quadratic polynomials in $\mathbb{Z}_p$

TL;DR

This paper investigates the structure of critical orbits for quadratic polynomials over p-adic integers, introducing a non-Archimedean analogue of Hubbard trees and analyzing specific cases like $c \, \in \, \mathbb{Z}_3$.

Contribution

It proposes a new notion of Hubbard trees in the non-Archimedean setting and characterizes possible structures for quadratic polynomials over $\, \mathbb{Z}_p$.

Findings

Classifies Hubbard tree structures in the p-adic context

Provides detailed analysis for $f_c$ with $c \, \in \, \mathbb{Z}_3$

Establishes a framework for understanding p-adic polynomial dynamics

Abstract

We study the forward orbit of the critical point for polynomials of the form $f_c=z^2+c$ defined over $\mathbb{Z}_p$. Hubbard trees capture the dynamical behavior for such maps with finite critical orbit in $\mathbb{C}$. We suggest a notion of Hubbard trees in the non-Archimedean setting, and describe the possible structures that arise for polynomials in $\mathbb{Z}_p$. As an example, we take a closer look at the dynamics of $f_c$ for $c\in\mathbb{Z}_3$.