1.Basic Introduction

Let

$$p(x)=\sum _{i=0}^{\mathrm{\infty }}{C}_i{x}^i$$

with $C_i\in \mathbb{R},\mathrm{\forall }i\in \mathbb{N}$. Let

$$p_j(x)=\sum _{i=1}^{\mathrm{\infty }}{C}_{i,j}{x}^i,$$

the $j$th composition of $p$. We know that $p_j$ exists because functions stay analytic under composition. Even if the original function did not converge, $p_j$ still has a meaning as a formal serie. Some trivial properties: + $p_0(x)=x$ + $p_1(x)=p(x)$ The case with $C_0\ne 0$ is complicated as all coeficient have an impact on all coeficients. We will thus focus on the case $C_0=0$