Dynamical diophantine approximation exponents in characteristic p $p$

Let $\varphi \left(z\right)$ be a non-isotrivial rational function in one-variable with coefficients in ${\overline{F}}_p\left(t\right)$ and assume that $\gamma \in P^1\left({\overline{F}}_p\left(t\right)\right)$ is not a post-critical point for $\varphi$. Then we prove that the diophantine approximation exponent of elements of ${\varphi }^{-m}\left(\gamma \right)$ are eventually bounded above by $\lceil deg{\left(\varphi \right)}^m/2\rceil +1$. 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 $a\in P^1\left({\overline{F}}_p\left(t\right)\right)$ and write ${\varphi }^n\left(a\right)=a_n/b_n$ for some coprime polynomials $a_n,b_n\in {\overline{F}}_p\left[t\right]$, then we prove that