The Fatou coordinate for parabolic Dulac germs

Abstract

We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated logarithm monomials. Acknowledgement. This research was supported by: Croatian Science Foundation (HRZZ) project no. 2285, French ANR project STAAVF, French-Croatian bilateral Cogito project 33003TJ Classification de points fixes et de singularités à l'aide d'epsilon-voisinages d'orbites et de courbes, Croatian UKF project Classifications of Dulac maps and epsilon-neighborhoods 2018, the University of Zagreb research support for 2015 and 2016.

Realization of analytic moduli for parabolic Dulac germs

“…We prove (A.2) using (A.3) proven in step 1. We repeat the construction of the Fatou coordinates for f on petals, described in detail in [9] and in [7, §8], but deducing the uniform bounds. Consider the Abel equation for f :…”

“…The definition of a transserial asymptotic expansion of a certain type is dependent on the choice of the summation method at limit ordinal steps. This choice is called a section function in [9]. To ensure uniqueness of the asymptotic expansion, we should be able to make a canonical choice of the section function.…”

“…To ensure uniqueness of the asymptotic expansion, we should be able to make a canonical choice of the section function. See [9] for more details on the problem of well-defined transserial asymptotic expansions and the notion of section functions.…”

“…For technical reasons in the Cauchy-Heine construction, on the standard quadratic domain we get realization results by germs for which we are unable to prove unicity of the transserial asymptotic expansion after the first three terms. To be able to define the unique transserial asymptotic expansion of a germ of a certain type, we should be able to prescribe a canonical method of summation, or section function [9], at limit ordinal steps. In the linear case, the estimates of the Cauchy-Heine integrals give us sufficiently good Gevrey-like bounds, and thus a canonical way to attribute the sum, at limit ordinal steps.…”

“…Here, = 1/(−log z). As discussed in [9], an asymptotic expansion of a germ in L(R) is in general neither well defined nor unique. The generalized Dulac expansion is a sectional asymptotic expansion (see [9] for precise definition of sections) that becomes unique after a canonical choice of section functions (the summation method) at limit ordinal steps-here, the log-Gevrey sums of a certain order.…”