abstract: This research is motivated by the question of cyclicity for hyperbolic polycycles in planar vector fields. We consider one-dimensional germs (at fixed point 0) which admit asymptotic expansion in power-logarithmic scale (the Dulac maps). We discuss embedding of such maps in a flow as time-one maps, that is, their rectifying Fatou coordinate. We study the transserial nature of an asymptotic expansion of the Fatou coordinate, and define an appropriate notion of integral asymptotic expansions to ensure uniqueness of the expansion. Finally, we motivate our work by fractal analysis: we answer the question of reading the formal class of a Dulac germ from the initial part of the expansion of the length of the epsilon-neighborhood of only one orbit.
This is a joint work with P. Mardešić, J.P. Rolin and V. Županović.