Dynamical diophantine approximation exponents in characteristic p

Let phi(z) be a non-isotrivial rational function in one-variable with coefficients in (F) over bar (p)(t) and assume that gamma is an element of P1((F) over bar (p)(t)) is not a post-critical point for phi. Then we prove that the diophantine approximation exponent of elements of phi(-m) are eventually bounded above by inverted right perpendicular deg(phi)(m)/2inverted left perpendicular + 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 is an element of P-1((F) over bar (p)(t)) and write phi(n)(a)=a(n)/b(n) for some coprime polynomials a(n), b(n) is an element of(F) over bar (p)[t], then we prove that 1/2 <= lim inf(n ->infinity)deg(a(n))/deg(b(n)) <= lim sup(n ->infinity)deg(a(n))/deg(b(n)) <= 2, whenever 0 and infinity 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.