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Realization of analytic moduli for parabolic Dulac germs

Part of: Qualitative theory Smooth dynamical systems: general theory Complex dynamical systems

Published online by Cambridge University Press:  18 January 2021

PAVAO MARDEŠIĆ Open the ORCID record for PAVAO MARDEŠIĆ [Opens in a new window]  and
MAJA RESMAN
PAVAO MARDEŠIĆ
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, F-21000Dijon, France
MAJA RESMAN
Affiliation:
University of Zagreb, Faculty of Science, Department of Mathematics, Bijenička 30, 10000Zagreb, Croatia
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Abstract

In a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli.

Keywords

almost regular germsanalytic invariantsEcalle–Voronin moduliGevrey expansionsCauchy–Heine integrals

MSC classification

Primary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37F75: Holomorphic foliations and vector fields
Secondary: 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 195 - 249
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Ahlfors, L.. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill, New York, 1979.Google Scholar
Ilyashenko, Y.. Finiteness theorems for limit cycles. Russian Math. Surveys 45(2) (1990), 143–200; Finiteness theorems for limit cycles. Trans. Amer. Math. Soc. 94 (1991).Google Scholar
Ilyashenko, Y. and Yakovenko, S.. Lectures on Analytic Differential Equations (Graduate Studies in Mathematics, 86). American Mathematical Society, Providence, RI, 2008.Google Scholar
Loday-Richaud, M.. Divergent Series, Summability and Resurgence II: Simple and Multiple Summability (Lecture Notes in Mathematics). Springer, Cham, 2016.CrossRefGoogle Scholar
Loray, F.. Analyse des séries divergentes. Quelques aspects des mathématiques actuelles. Ellipses, Paris, 1998.Google Scholar
Loray, F.. Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. Prépublication IRMAR, ccsd-00016434, 2005.Google Scholar
Mardešić, P. and Resman, M.. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.CrossRefGoogle Scholar
Mardešić, P., Resman, M., Rolin, J. P. and Županović, V.. Normal forms and embeddings for power-log transseries. Adv. Math. 303 (2016), 888–953.CrossRefGoogle Scholar
Mardešić, P., Resman, M., Rolin, J. P. and Županović, V.. The Fatou coordinate for parabolic Dulac germs. J. Differential Equations 266(6) (2019), 3479–3513.CrossRefGoogle Scholar
Roussarie, R.. Bifurcations of Planar Vector Fields and Hilbert’s Sixteenth Problem. Birkhäuser, Basel, 1998.CrossRefGoogle Scholar
Schäfke, R. and Teyssier, L.. Analytic normal forms for convergent saddle-node vector fields. Ann. Inst. Fourier (Grenoble) 65(3) (2015), 933–974.CrossRefGoogle Scholar
Voronin, S. M.. Analytic classification of germs of conformal mappings $\left(\mathbb{C},0\right)\to \left(\mathbb{C},0\right)$ . Funktsional. Anal. i Prilozhen. 15(1) (1981), 1–17 (in Russian).CrossRefGoogle Scholar