Computing Dulac's maps of (almost every) non-degenerate singularities

Loïc Jean Dit Teyssier (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We study the complex Dulac map for a holomorphic foliation near a non-degenerate singularity (both eigenvalues of the linearization are nonzero). We describe the order of magnitude of the first two terms of the asymptotic expansion and show how to compute explicitly those terms using characteristics supported in the leaves of the linearized foliation. We perform similarly the study of the Dulac time spent around the singularity. These 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 arithmetics of the eigenvalues) provided the foliation has enough (i.e. two) separatrices. The study aims at being as explicit as possible, in particular by giving as precise a bound as possible on the remainder in the asymptotic expansion.

Domaines: Systèmes dynamiques [math.DS]

Loïc Jean Dit Teyssier. Computing Dulac's maps of (almost every) non-degenerate singularities. 2012. ⟨hal-00771523v1⟩