Explicit linearization of multi-dimensional germs and vector fields through Ecalle's tree expansions

TL;DR

This paper introduces explicit, non-recursive formulas for linearizing transformations of non-resonant analytic germs of diffeomorphisms and vector fields in complex dimensions, using Ecalle's tree-based combinatorics, and provides convergence estimates under Bruno's condition.

Contribution

It offers a novel explicit formula for linearization transformations based on Ecalle's armould calculus, applicable to high-dimensional systems, with improved convergence estimates.

Findings

Explicit formulas for linearization transformations in any complex dimension.

Recovery of optimal convergence estimates under Bruno's condition.

New dependence of convergence domains on system dimension.

Abstract

We provide explicit formulas of non-recursive type for the linearizing transformations of a non-resonant analytic germ of diffeomorphism at a fixed point or a non-resonant analytic germ of vector field at a singular point, in any complex dimension. The formal expressions we obtain rely on a part of Ecalle's tree-based combinatorics called ``armould calculus" and they have the same shape for dynamical systems with discrete or continuous time. They allow us to recover in a straightforward manner, under Bruno's arithmetical condition, the best known estimates for the domains of convergence of the analytic 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.