On iterations of rational functions over perfect fields

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.