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.

18 pages; comments are welcome

References in corpus (2)

• Irreducible polynomials over finite fields produced by composition of quadratics
• Construction of irreducible polynomials through rational transformations