For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$ Deduce the Block and Thielmann theorem: if $(F_n)_{n \in \mathbb{N}}$ satisfies (III.1), then there exists $U \in G$ such that $$\forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ X^n \circ U \quad \text{or} \quad \forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ T_n \circ U$$
For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying
$$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Deduce the Block and Thielmann theorem: if $(F_n)_{n \in \mathbb{N}}$ satisfies (III.1), then there exists $U \in G$ such that
$$\forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ X^n \circ U \quad \text{or} \quad \forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ T_n \circ U$$