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Tubular neighborhoods of orbits of power-logarithmic germs

  • P. MardešićEmail author
  • M. Resman
  • J.-P. Rolin
  • V. Županović
Article
  • 34 Downloads

Abstract

We consider a class of power-logarithmic germs. We call them parabolic Dulac germs, as they appear as Dulac germs (first-return germs) of hyperbolic polycycles. In view of formal or analytic characterization of such a germ f by fractal properties of several of its orbits, we study the tubular \(\varepsilon \)-neighborhoods of orbits of f with initial points \(x_0\). We denote by \(A_f(x_0,\varepsilon )\) the length of such a tubular \(\varepsilon \)-neighborhood. We show that, even if f is an analytic germ, the function \(\varepsilon \mapsto A_f(x_0,\varepsilon )\) does not have a full asymptotic expansion in \(\varepsilon \) in the scale of powers and (iterated) logarithms. Hence, this partial asymptotic expansion cannot contain necessary information for analytic classification. In order to overcome this problem, we introduce a new notion: the continuous time length of the\(\varepsilon \)-neighborhood\(A^c_f(x_0,\varepsilon )\). We show that this function has a full transasymptotic expansion in \(\varepsilon \) in the power, iterated logarithm scale. Moreover, its asymptotic expansion extends the initial, existing part of the asymptotic expansion of the classical length \(\varepsilon \mapsto A_f(x_0,\varepsilon )\). Finally, we prove that this initial part of the asymptotic expansion determines the class of formal conjugacy of the Dulac germ f.

Keywords

Dulac map Fractal properties of orbits \(\varepsilon \)-Neighborhoods Power-logarithm asymptotic expansions Transseries Formal and analytic invariants Embedding in a flow 

Mathematics Subject Classification

28A75 37C05 37C15 34C07 30B10 34E05 37C10 

Notes

Acknowledgements

This research was supported by: Croatian Science Foundation (HRZZ) Grant Nos. 2285, UIP 2017-05-1020, PZS-2019-02-3055 from “Research Cooperability” funded by the European Social Fund, Croatian Unity Through Knowledge Fund (UKF) My first collaboration grant Grant No. 7, French ANR project STAAVF. Part of the research was made during the 6-month stay of M. Resman at Université de Bourgogne in 2018, financed by the Croatian UKF project.

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Authors and Affiliations

  1. 1.Institut de Mathématiques de Bourgogne, UMR 5584, CNRSUniversité Bourgogne Franche-ComtéDijonFrance
  2. 2.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia
  3. 3.Department of Applied Mathematics, Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia

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