1.Basic Introduction

Let

\[ p(x)=\sum_{i=0}^{\infty}C_ix^i \]

with \(C_i\in\mathbb{R}, \forall i \in \mathbb{N}\). Let

\[ p_j(x)=\sum_{i=1}^\infty C_{i,j}x^i, \]

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 \(C_0=0\)