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My question is an amalgamation of two previous questions. The first question I'd like to draw attention to is here. It asks whether there can exist a non trivial semigroup defined on $\mathbb{C}$ $$\...

Assume that $f(z)$ is a holomorphic function that sends some open and connected set $G$ to itself. Assume $f$ has a single fixed point $z_0$. Assume $f(f(...(n\,times)...f(z))) = f^{\circ n}(z) \to ...