Extending to the complex line Dulac's corner maps of non-degenerate planar singularities

TL;DR

This paper provides a comprehensive geometric analysis of the complex Dulac map near non-degenerate singularities in holomorphic foliations, detailing its domain, boundary regularity, asymptotic expansion, and Dulac time, without restrictions on linearizability or eigenvalue ratios.

Contribution

It introduces a unified geometric framework to study the Dulac map's domain, boundary, asymptotics, and Dulac time for non-degenerate singularities, independent of linearizability or eigenvalue ratios.

Findings

Complete description of the Dulac map's topology and boundary regularity.

Explicit bounds on the asymptotic expansion and remainder of the Dulac map.

Unified approach applicable regardless of linearizability and eigenvalue ratio.

Abstract

We study the complex Dulac map for a holomorphic foliation of the complex plane, near a non-degenerate singularity (both eigenvalues of the linearization are nonzero) with two separatrices. Following the well-known results of Y. Il'yashenko we provide a geometric approach allowing to study the whole maximal domain of (geometric) definition of the Dulac map. In particular its topology and the regularity of its boundary are completely described. We also study the order of magnitude of the first non-trivial term of its asymptotic expansion and show how to compute it using path integrals supported in the leaves of the linearized foliation. Explicit bounds on the remainder are given. We perform similarly the study of the Dulac time spent around the singularity. All results are formulated in a unified framework taking no heed to the usual dynamical discrimination (i.e. no matter whether the singularity is formally orbitally linearizable or not and regardless of the arithmetic of the eigenvalues ratio).