Universal uniqueness criterion?

[bo198214]

I think if both initial regions belong to the same fixed point pair, then both slogs are analytic continuations of each other, i.e. basically the same holomorphic function. But this is not yet proven.

I'll try to prove that this summer. just give me all the relevant facts.

ok, so we compete

two Abel functions: A1(f1(x)) = A1(x) + 1 , A2(f2(x)) = A2(x) + 1
you mean two different abel functions, (1) where f1, f2 are different functions? or do A1, A2 map the same point to different values? (for sake of argument, a "superlogarithm" that does slog(0) = 0)

two Abel functions A1, A2 of the same function f
A1(f(x))=A(x)+1, A2(f(x))=A2(x)+1, A1(d)=A2(d)=c.
but A1 is defined on the initial region G1 and A2 is defined on the initial region G2.
Of course \( d\in G_1\cap G_2\neq \emptyset \).