Length of epsilon-neighborhoods of orbits of Dulac maps

2016/06/08 by Pavao Mardešić, Mardesic, P., Maja Resman +5
Mathematics · #26A12 #28A75 #34C20 #37C10 #39B12 #46A19 #58K50 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1606.02581

openalex publication_date 2016/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 continuous time length of ε-neighborhoods of orbits, and to prove that this function of ε 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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