# Lie algebras and geometric complexity of an isochronous center condition

## TL;DR

This paper explores the geometric complexity of isochronous center conditions using mould formalism and Lie algebra structures, advancing the understanding of linearisability in dynamical systems.

## Contribution

It introduces a novel approach combining mould formalism and Lie ideals to analyze the geometric complexity of isochronous centers, extending previous theoretical frameworks.

## Key findings

- Identifies the role of Lie ideals in the geometric complexity analysis.
- Connects mould formalism with Lie algebra structures in the context of isochronous centers.
- Provides insights into the size and splitting of Lie ideals related to linearisability.

## Abstract

Using the mould formalism introduced by Jean Ecalle, we define and study the geometric complexity of an isochronous center condition. The role played by several Lie ideals is discussed coming from the interplay between the universal mould of the correction and the different Lie algebras generated by the comoulds. This strategy enters in the general program proposed by J. Ecalle and D. Schlomiuk in \cite{es} to study the size and splitting of some Lie ideals for the linearisability problem.