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The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects. (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