Explicit linearization of multi-dimensional germs and vector fields through Ecalle's tree expansions
We revisit the Siegel linearization problem by means of explicit formulas of non-recursive type for the linearizing transformations of a non-resonant holomorphic germ of diffeomorphism at a fixed point, or a non-resonant holomorphic germ of vector field at a singular point, in any complex dimension. These formal expressions, that have the same shape for dynamical systems with discrete or continuous time and were obtained long ago by J. Ecalle through his "arborification" technique, are derived here by a more direct application of his tree-based combinatorics called "armould calculus". They involve a certain family of differential operators indexed by decorated trees and forests. A combinatorial study of these differential operators allows us to recover, under Bruno's arithmetical condition, the best known estimates for the domains of convergence of the holomorphic linearizing changes of variables in terms of the value of the Bruno series, including a new precise dependence with respect to the dimension of the problem.