Universal uniqueness criterion?

[bo198214]

Haven't you already proved that for superlogarithms? ("for "each" simple intial region G (if both "boundaries", \( \partial_1 G \) and \( \partial_2 G \), are bounded by the fixed points of F, wouldn't that satisfy the criteria for being a simple region?) ... there exists at most one holomorphic Abel function for F...")

No, the uniqueness depends on the initial region G.
If we have two Abel functions biholomorphic on two different initial regions (even if those different initial regions belong to the same fixed point pair) the proposition is not applicable.