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Let K be a function field, let f be a rational function of degree d at least2 defined over K, and suppose that f is not isotrivial. In this paper, we showthat a point P in P^1(Kbar) has f-canonical height zero if and only if P ispreperiodic for f. This answers affirmatively a question of Szpiro and Tucker,and generalizes a recent result of Benedetto from polynomials to rationalfunctions. We actually prove the following stronger result, which is a variantof the Northcott finiteness principle: there exists epsilon > 0 such that theset of points P in P^1(K) with f-canonical height at most epsilon is finite.Our proof is essentially analytic, making use of potential theory on Berkovichspaces to prove some new results about the dynamical Green's functions g_v(x,y)attached to f at each place v of K. For example, we show that every conjugateof f has bad reduction at v if and only if g_v(x,x) > 0 for all x in theBerkovich projective line over the completion of the algebraic closure of K_v.In an appendix, we show how a similar method can be used to give a new proof ofthe Mordell-Weil theorem for elliptic curves over K.

A finiteness theorem for canonical heights attached to rational maps over function fields