Hey Ciera, welcome on board,

before my retreat, some points about your post:

I tried a bit about this "boundedness criterion" whether quasi-holomorphic or only holomorphic in the second variable. However I couldnt find a uniqueness criterion (for general functions not only exponentiation though).

Such a criterion would be of great value, because it would unite the case b<e^(1/e) with the case b>e^(1/e) because it is applicable for both.

I can give you only the two references containing uniqueness criterion for the case with real fixed point:

http://citeseerx.ist.psu.edu/viewdoc/sum...1.154.3249

and the case of two complex fixed points:

http://arxiv.org/abs/1006.3981

also Mike3 posted lately a nice uniqueness criterion for regular iteration at a real fixed point.

I am not sure whether considering the bivariate holomorphism really helps for uniqueness. But I wish you luck in proving such a criterion, really!

(Dont forget however to make sure that your uniqueness criterion can be satisfied by any function at all!)

before my retreat, some points about your post:

I tried a bit about this "boundedness criterion" whether quasi-holomorphic or only holomorphic in the second variable. However I couldnt find a uniqueness criterion (for general functions not only exponentiation though).

Such a criterion would be of great value, because it would unite the case b<e^(1/e) with the case b>e^(1/e) because it is applicable for both.

I can give you only the two references containing uniqueness criterion for the case with real fixed point:

http://citeseerx.ist.psu.edu/viewdoc/sum...1.154.3249

and the case of two complex fixed points:

http://arxiv.org/abs/1006.3981

also Mike3 posted lately a nice uniqueness criterion for regular iteration at a real fixed point.

I am not sure whether considering the bivariate holomorphism really helps for uniqueness. But I wish you luck in proving such a criterion, really!

(Dont forget however to make sure that your uniqueness criterion can be satisfied by any function at all!)