Equidistribution of small points, rational dynamics, and potential theory

Given a rational function phi(T) on P-1 of degree at least 2 with coefficients in a number field k, we show that for each place nu of k, there is a unique probability measure mu(phi,nu) on the Berkovich space P-Berk, v(1)/C-nu such that if {z(n)} is a sequence of points in P-1((k) over bar) whose W-canonical heights tend to zero, then the z(n)'s and their Gal((k) over bar /k)-conjugates are equidistributed with respect to mu(phi, nu). The proof uses a polynomial lift F(x, y) = (F-1 (x, y), F-2 (x, y)) of phi to construct a two-variable Arakelov-Green's function g(phi, nu) (x, y) for each v. The measure mu(phi, nu) is obtained by taking the Berkovich space Laplacian of g(phi, nu) (x, y). The main ingredients in the proof are an energy minimization principle for g(phi, nu) (x, y) and a formula for the homogeneous transfinite diameter of the nu-adic filled Julia set K-F,K- nu subset of C-nu(2) for each place nu.