On iterations of rational functions over perfect fields

Let $\mathbb{K}$ be a perfect field of characterstic $p\ge 0$ and let $R\in \mathbb K(x)$bearationalfunction\mathrm{.}Thispaperstudiesthenumber$\Delta_{\alpha, R}(n)$ofdistinctsolutionsof$R^{(n)}(x)=\alpha$ over the algebraic closure $\bar{\mathbb{K}}$ of $\mathbb{K}$, where $\alpha\in \overline{\mathbb K}$and$R^{(n)}$isthe$n$-foldcompositionof$R$ with itself. With the exception of some pairs $(\alpha ,R)$, we prove that ${\mathrm{\Delta }}_{\alpha ,R}(n)=c_{\alpha ,R}\cdot d^n+O_{\alpha ,R}(1)$ 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 ${\mathrm{\Delta }}_{\alpha ,R}(n)\le 2$ 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.