  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.
