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## Cute Problem with functions

Let $A\subset \mathbb{C}$ and $f:A\rightarrow A$ A function. Define $f_1=f$ and $f_{k+1}=f_k\circ f,\ (\forall)k \in \mathbb{N}^*$. Assume that $(\exists) \alpha,\ \beta>0$ with $\alpha+\beta =1$ and $m,\ n\in \mathbb{N}^*$ coprimes ( $\gcd(m,n)=1$), such that $\alpha f_m(x)+\beta f_n(x)=x,\ (\forall)x\in A$. Find all functions $f$ in the following cases:

i) $A=\mathbb{N}$.

ii) $(\exists)a\in \mathbb{C}^*,\ p\in \mathbb{N}^*$ such that $A=\{z\in \mathbb{C}\ |\ z^p=a\}.$

iii) Any set $A\subset \mathbb{C}$ such that there are no three collinear points in $A$.