The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects (Q2484114)

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.