Dynamical Diophantine Approximation Exponents in Characteristic $p$

Let $\phi (z)$ be a non-isotrivial rational function in one-variable with coefficients in ${\bar{\mathbb{F}}}_p(t)$ and assume that $\gamma \in {\mathbb{P}}^1({\bar{\mathbb{F}}}_p(t))$ 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 $\lceil d^m\mathrm{/}2\rceil +1$. 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 $a\in {\mathbb{P}}^1({\bar{\mathbb{F}}}_p(t))$ and write ${\phi }^n(a)=a_n\mathrm{/}{b}_n$ for some coprime polynomials $a_n,b_n\in {\bar{\mathbb{F}}}_p[t]$, then we prove that $$\frac{1}{2}\le \underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\text{deg}(a_n)}{\text{deg}(b_n)}\le \underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\text{deg}(a_n)}{\text{deg}(b_n)}\le 2,$$ whenever $0$ and $\mathrm{\infty }$ 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.