  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.
