Length of epsilon-neighborhoods of orbits of Dulac maps

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.