OMG, I just found out Sheldon asked that same question already on MSE Jul 2013!
https://math.stackexchange.com/questions...-iteration
And he got an answer!
Maybe we should read
MR0729400 S. Dubuc, Étude théorique et numérique de la fonction de Karlin-McGregor, J. Analyse Math. 42 (1982/83), 15–37.
However I don't have access to university anymore, maybe JmsNxn can procuree this article.
However this article deals with an older article
Samuel Karlin and James McGregor, Embedding iterates of analytic functions with two fixed points into continuous groups, Trans. Amer. Math. Soc. 132 (196
, 137–145. MR 224790, DOI 10.1090/S0002-9947-1968-0224790-2
which is freely accessible here (I also append it to this post).
However when skimming through this article. He only considers a very particular group of holomorphic function that do not allow branches for example.
From this functions he concludes that the only ones where the iterations coincide at both fixed points are the linear fractional functions.
I think we already a step ahead in our discussion here
https://math.stackexchange.com/questions...-iteration
And he got an answer!
Maybe we should read
MR0729400 S. Dubuc, Étude théorique et numérique de la fonction de Karlin-McGregor, J. Analyse Math. 42 (1982/83), 15–37.
However I don't have access to university anymore, maybe JmsNxn can procuree this article.
However this article deals with an older article
Samuel Karlin and James McGregor, Embedding iterates of analytic functions with two fixed points into continuous groups, Trans. Amer. Math. Soc. 132 (196
, 137–145. MR 224790, DOI 10.1090/S0002-9947-1968-0224790-2which is freely accessible here (I also append it to this post).
However when skimming through this article. He only considers a very particular group of holomorphic function that do not allow branches for example.
From this functions he concludes that the only ones where the iterations coincide at both fixed points are the linear fractional functions.
I think we already a step ahead in our discussion here
