Eventually Stable Quadratic Polynomials over Q(i)

Abstract

Let $f$ be a polynomial or a rational function over a field $K$. Arithmetic dynamics studies the algebraic and number-theoretic properties of its iterates $f^n:=f \circ f \circ ... \circ f$.\ A basic question is, if $f$ is a polynomial, are these iterates irreducible or not? We wish to know what can happen when considering iterates of a quadratic $f= x^2+r\in K[x]$. The most interesting case is when $r=\frac{1}{c}$, which we will focus on, and discuss criteria for irreducibility, i.e. \emph{stability} of all iterates. We also wish to prove that if 0 is not periodic under $f$, then the number of factors of $f^n$ is bounded by a constant independent of $n$, i.e. $f$ is \emph{eventually stable}. This thesis is an extension to $\Qi$ of the paper \cite{evstb}, which considered $f$ over $\mathbb{Q}$. This thesis involves a mixture of ideas from number theory and arithmetic geometry. We also show how eventual stability of iterates ties into the density of prime divisors of sequences.