Dynamical Diophantine Approximation Exponents in Characteristic $p$

Abstract

Let $\varphi (z)$ be a non-isotrivial rational function in one-variable with coefficients in ${\overline{\mathbb{F}}}_p(t)$ and assume that $\gamma \in {\mathbb{P}}^1({\overline{\mathbb{F}}}_p(t))$ is not a post-critical point for $\varphi$. Then we prove that the diophantine approximation exponent of elements of ${\varphi }^{-m}(\gamma )$ are eventually bounded above by $\lceil d^m/2\rceil +1$. To do this, we mix diophantine techniques in characteristic $p$ with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we take any wandering point $a\in {\mathbb{P}}^1({\overline{\mathbb{F}}}_p(t))$ and write ${\varphi }^n(a)=a_n/b_n$ for some coprime polynomials $a_n,b_n\in {\overline{\mathbb{F}}}_p[t]$, then we prove that \[ \frac{1}{2}\leq \liminf_{n\rightarrow\infty} \frac{\text{deg}(a_n)}{\text{deg}(b_n)} \leq\limsup_{n\rightarrow\infty} \frac{\text{deg}(a_n)}{\text{deg}(b_n)}\leq2,\] whenever $0$ and $\infty$ are both not post-critical points for $\varphi$. In characteristic $p$, the Thue-Siegel-Dyson-Roth theorem is false, and so our proof requires different techniques than those used by Silverman.