On iterations of rational functions over perfect fields

TL;DR

This paper investigates the growth of the number of solutions to iterated rational functions over perfect fields, establishing exponential bounds and characterizing exceptional cases with limited solutions.

Contribution

It provides a general formula for the solution count of iterated rational functions over perfect fields, identifying exceptional cases and offering bounds on solution numbers.

Findings

Solution count grows exponentially with iteration number

Exceptional pairs have at most two solutions for all iterations

Explicit bounds and characterizations of exceptional pairs

Abstract

Let $\mathbb K$ be a perfect field of characterstic $p\ge 0$ and let $R\in \mathbb K(x)$ be a rational function. This paper studies the number $\Delta_{\alpha, R}(n)$ of distinct solutions of $R^{(n)}(x)=\alpha$ over the algebraic closure $\overline{\mathbb K}$ of $\mathbb K$, where $\alpha\in \overline{\mathbb K}$ and $R^{(n)}$ is the $n$-fold composition of $R$ with itself. With the exception of some pairs $(\alpha, R)$, we prove that $\Delta_{\alpha, R}(n)=c_{\alpha, R}\cdot d^n+O_{\alpha, R}(1)$ for some $0<c_{\alpha, R}\le 1<d$. The number $d$ is readily obtained from $R$ and we provide estimates on $c_{\alpha, R}$. Moreover we prove that the exceptional pairs $(\alpha, R)$ satisfy $\Delta_{\alpha, R}(n)\le 2$ for every $n\ge 0$, and we fully describe them. We also discuss further questions and propose some problems in the case where $\mathbb K$ is finite.