Dynamical diophantine approximation exponents in characteristic p$p$

Abstract Let be a non‐isotrivial rational function in one‐variable with coefficients in and assume that is not a post‐critical point for . Then we prove that the diophantine approximation exponent of elements of are eventually bounded above by . To do this, we mix diophantine techniques in characteristic 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 and write for some coprime polynomials , then we prove that whenever 0 and are both not post‐critical points for . In characteristic , the Thue–Siegel–Dyson–Roth theorem is false, and so our proof requires different techniques than those used by Silverman.