Combinatorics of Poincar\'e's and Schr\"oder's equations

TL;DR

This paper explores the combinatorial structure of Poincaré's and Schröder's equations through symmetric functions, noncommutative algebra, and operad theory, revealing new algebraic and combinatorial insights.

Contribution

It introduces a novel interpretation of the conjugacy equation using symmetric functions, noncommutative symmetric functions, and operad theory, connecting classical equations to modern algebraic structures.

Findings

Explicit expansion of solutions in terms of plane trees

Coefficients in the ribbon basis are in ] after normalization

Connection between the conjugacy equation and operad of Stasheff polytopes

Abstract

We investigate the combinatorial properties of the functional equation $\phi[h(z)]=h(qz)$ for the conjugation of a formal diffeomorphism $\phi$ of $\mathbb{C}$ to its linear part $z\mapsto qz$. This is done by interpreting the functional equation in terms of symmetric functions, and then lifting it to noncommutative symmetric functions. We describe explicitly the expansion of the solution in terms of plane trees and prove that its expression on the ribbon basis has coefficients in ${\mathbb N}[q]$ after clearing the denominators $(q)_n$. We show that the conjugacy equation can be lifted to a quadratic fixed point equation in the free triduplicial algebra on one generator. This can be regarded as a $q$-deformation of the duplicial interpretation of the noncommutative Lagrange inversion formula. Finally, these calculations are interpreted in terms of the group of the operad of Stasheff polytopes, and are related to Ecalle's arborified expansion by means of morphisms between various Hopf algebras of trees.