Dynamical Diophantine Approximation Exponents in Characteristic $p$

Let $\phi(z)$ 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 $\phi$. Then we prove that the diophantine approximation exponent of elements of $\phi^{-m}(\gamma)$ 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 $\phi^n(a)=a_n/b_n$ 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 $\phi$. In characteristic $p$, the Thue-Siegel-Dyson-Roth theorem is false, and so our proof requires different techniques than those used by Silverman.