Variation of canonical height for\break Fatou points on P-1

Let f W P-1 -> P-1 be a map of degree > 1 defined over a function field k = K(X), where K is a number field and X is a projective curve over K. For each point a is an element of P-1(k) satisfying a dynamical stability condition, we prove that the Call-Silverman canonical height for specialization f(t) at point a(t), for t is an element of X((Q) over bar) outside a finite set, induces a Weil height on the curve X; i.e., we prove the existence of a Q-divisor D = D-f,D- a on X so that the function t sec -> (h) over capf(t).(a(t)) - h(D)(t) is bounded on X((Q) over bar) for any choice of Weil height associated to D. We also prove a local version, that the local canonical heights t sec -> (lambda) over cap f(t),v(at) differ from a Weil function for D by a continuous function on X(C-v), at each place v of the number field K. These results were known for polynomial maps f and all points a is an element of P-1(k)without the stability hypothesis, and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a is an element of P-1(k). Finally, we characterize our stability condition in terms of the geometry of the induced map sec f X x P-1 over K over K; and we prove the existence of relative Neron models for the pairsecf: (f,a)when a is a Fatou point at a place gamma of k, where the local canonical height (lambda) over cap (f,gamma)(a) can be computed as an intersection number.