Dynamical Diophantine Approximation Exponents in Characteristic $p$

Abstract

Let φ(z) be a non-isotrivial rational function in one-variable with coefficients in Fp(t) and assume that γ ∈ P 1 (Fp(t)) is not a post-critical point for φ. Then we prove that the diophantine approximation exponent of elements of φ −m (γ) are eventually bounded above by ⌈d m /2⌉ + 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 ∈ P 1 (Fp(t)) and write φ n (a) = an/bn for some coprime polynomials an, bn ∈ Fp[t], then we prove thatwhenever 0 and ∞ are both not post-critical points for φ. In characteristic p, the Thue-Siegel-Dyson-Roth theorem is false, and so our proof requires different techniques than those used by Silverman.