Equidistribution of small points, rational dynamics, and potential theory

Abstract

If phi(z) is a rational function on P^1 of degree at least 2 with coefficients in a number field k, we compute the homogeneous transfinite diameter of the v-adic filled Julia sets of phi for all places v of k by introducing a new quantity called the homogeneous sectional capacity. In particular, we show that the product over all places of these homogeneous transfinite diameters is 1. We apply this product formula and some new potential-theoretic results concerning Green's functions on Riemann surfaces and Berkovich spaces to prove an adelic equidistribution theorem for dynamical systems on the projective line. This theorem, which generalizes the results of Baker-Hsia, says that for each place v of k, there is a canonical probability measure on the Berkovich space P^1_{Berk,v} over C_v such that if z_n is a sequence of algebraic points in P^1 whose canonical heights with respect to phi tend to zero, then the z_n's and their Galois conjugates are equidistributed with respect to mu_{phi,v} for all places v of k. For archimedean v, P^1_{Berk,v} is just the Riemann sphere, mu_{phi,v} is Lyubich's invariant measure, and our result is closely related to a theorem of Lyubich and Freire-Lopes-Mane.