Length of epsilon-neighborhoods of orbits of Dulac maps

P. Mardesic, M. Resman, J. -P. Rolin, V. Zupanovic

Published 2016-06-08Version 1

Dulac maps are first return maps of hyperbolic polycycles of analytic planar vector fields. We study the fractal properties of their orbits. For this purpose, we introduce a new notion, the \emph{continuous time length of $\varepsilon$-neighborhoods of the orbits}, and prove that this function of $\varepsilon$ admits an asympotic expansion in terms of transseries. Given a parabolic Dulac germ, we prove that this expansion determines its class of formal conjugacy, and compute its Fatou coordinate.