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arXiv · 1606.02581

Length of epsilon-neighborhoods of orbits of Dulac maps

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

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P. Mardesic, M. Resman, J. -P. Rolin, V. Zupanovic. 2016-06-08. Length of epsilon-neighborhoods of orbits of Dulac maps. https://arxiv.org/abs/1606.02581

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