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Zbl 1078.37016
Ecalle, Jean; Vallet, Bruno
The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects. (English)
[J] Ann. Fac. Sci. Toulouse, VI. Sér., Math. 13, No. 4, 575-657 (2004). ISSN 0240-2963

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-conver\-gent infinite series of type $\sum A^\omega B_\omega$ (called mould-comould expansion) the arborification/co\-arbor\-i\-fi\-cation 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.
[Filippo Bracci (Roma)]

MSC 2000:: *37C99 Smooth dynamical systems
34M15 Algebraic aspects of differential equations in the complex domain
37E20 Universality, renormalization
37F50 Small divisors, rotation domains and linearization
37G05 Normal forms
40A05 Convergence of series and sequences
32S65 Singularities of holomorphic vector fields

Keywords: non-convergent series; resummation; holomorphic dynamics; small divisors; resonances

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