Explicit linearization of one-dimensional germs through tree-expansions

Abstract

We explainÉcalle's "arbomould formalism" in its simplest instance, showing how it allows one to give explicit formulas for the operators naturally attached to a germ of holomorphic map in one dimension. When applied to the classical linearization problem of non-resonant germs, which contains the well-known difficulties due to the so-called small divisor phenomenon, this elegant and concise tree formalism yields compact formulas, from which one easily recovers the classical analytical results of convergence of the solution under suitable arithmetical conditions on the multiplier. We rediscover this way Yoccoz's lower bound for the radius of convergence of the linearization and can even reach a global regularity result with respect to the multiplier (C 1-holomorphy) which improves on Carminati-Marmi's result.

Resurgent Functions and Alien Calculus

“…In some cases, one can even reach analytic conclusions by means of the Écalle's 'arborification' technique-see [FM17] for connections between arborified moulds and the Connes-Kreimer Hopf algebras of trees and graphs and their applications to perturbative quantum field theory. Other examples of application of mould calculus to formal or analytic dynamical problems can be found in [ Éca92; ÉV98], [PS17], [Nov+18], [FMS18].…”