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Analytic linearization of hyperbolic (complex) Dulac germs

Dulac germs are analytic germs defined on special subdomains of the Riemann surface of the logarithm called standard quadratic domains. Furthermore, they admit certain logarithmic asymptotic expansions at zero called Dulac series. The first return maps of hyperbolic polycycles of analytic planar vector fields turn to be Dulac germs, which relates them to the Dulac problem of non- accumulation of limit cycles on a hyperbolic or semi-hyperbolic polycycle of an analytic planar vector field, solved independently by Ilyashenko [2] and Écalle [1]. First, we consider analytic maps on certain invariant complex domains with logarithmic asymptotic bounds. We present sufficient conditions for such maps to be analytically linearized [3]. Afterwards, we present the formal linearization result for hyperbolic Dulac series with complex coefficients [4] and define complex Dulac germs (a generalization of Dulac germs with complex coefficients in their asymptotic expansions at zero, [3]). We apply these formal and analytic results on the class of all hyperbolic (complex) Dulac germs in order to prove the linearization theorem [3] for hyperbolic (complex) Dulac germs, which is the main goal of this talk. References: J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques, Hermann, Paris, 1992. Y. Il'yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, vol.94, American Mathematical Society, Providence, RI, 1991. D. Peran, M. Resman, J. P. Rolin and T. Servi, Linearization of complex hyperbolic Dulac germs, Journal of Mathematical Analysis and Applications, 508(1), 1-27, 2022, https://doi.org/10.1016/j.jmaa.2021.125833 D. Peran, M. Resman, J. P. Rolin and T. Servi, Normal forms of hyperbolic logarithmic transseries, submitted, 2021. https://arxiv.org/pdf/2105.10660.pdf

analytic linearization, (complex) Dulac germs, Dulac series, standard quadratic domains, iteration theory, Koenigs sequence

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