Length of epsilon-neighborhoods of orbits of Dulac maps

TL;DR

This paper investigates the fractal properties of orbits of parabolic Dulac maps, introducing a new concept of continuous time length of epsilon-neighborhoods and linking it to the formal conjugacy class of the map.

Contribution

It establishes the existence of a Fatou coordinate with an asymptotic expansion for Dulac maps and introduces the continuous time length of epsilon-neighborhoods, connecting it to formal conjugacy.

Findings

The epsilon-neighborhood length admits an asymptotic expansion.

The expansion determines the formal conjugacy class.

A Fatou coordinate with a power-iterated logarithm asymptotic expansion exists.

Abstract

By Dulac maps we mean first return maps of hyperbolic polycycles of analytic planar vector fields. We study the fractal properties of the orbits of a parabolic Dulac map. To this end, we prove that it admits a Fatou coordinate with an asympotic expansion in terms of power-iterated logarithm transseries. This allows to introduce a new notion, the \emph{continuous time length of $\varepsilon$-neighborhoods of orbits}, and to prove that this function of $\varepsilon$ admits an asymptotic expansion in the same scale. We show that, under some hypotheses, this expansion determines the class of formal conjugacy of the Dulac map.