*(English)*Zbl 1078.37016

This paper is a long and detailed survey (but presenting also some new formulary) on the so-called arborification/coarborification transforms. Very roughly speaking, taken a non-convergent infinite series of type $\sum {A}^{\omega}{B}_{\omega}$ (called mould-comould expansion) the arborification/coarborification transform is a way of properly selecting indices to be summed in the mould part ${A}^{\omega}$ and in the comould part ${B}_{\omega}$ in order to obtain (at least in most of the interesting cases) a convergent series.

In the paper under review, the authors try to explain both combinatorial and algebraic aspects of the arborification/coarborification process and give several applications of this technique to analysis. For instance, just to name a few, they apply this process to the linearization of vector fields and diffeomorphisms with diophantine or resonant spectra and to KAM theory.

The paper is well organized and each part contains a preamble with heuristic and understandable explanations of what comes. On the other hand, the mathematical part itself might be a little hard for non-experts because of the systematic use of non-standard notations which is not always explained in the present paper.

##### MSC:

37C99 | Smooth dynamical systems |

34M15 | Algebraic aspects of ODE in the complex domain |

37E20 | Universality, renormalization |

37F50 | Small divisors, rotation domains and linearization; Fatou and Julia sets |

37G05 | Normal forms |

40A05 | Convergence and divergence of series and sequences |

32S65 | Singularities of holomorphic vector fields and foliations |