Formal normal forms and formal embeddings into flows for power-log transseries

TL;DR

This paper develops formal normal forms and embedding theorems for power-log transseries, extending Dulac series, to better understand their structure and behavior near hyperbolic polycycles.

Contribution

It introduces new algebraic frameworks for power-log transseries and establishes formal normalization and embedding results within these algebras.

Findings

Established formal normal forms for power-log transseries.

Proved formal embedding theorems for these transseries.

Extended Dulac series to more general algebraic structures.

Abstract

The Dulac series are the asymptotic expansions of first return maps in a neighborhood of a hyperbolic polycycle. In this article, we consider two algebras and of power-log transseries (generalized series) which extend the algebra of Dulac series. We give a formal normal form and prove a formal embedding theorem for transseries in these algebras.