Naslov
Analytic normalization of strongly hyperbolic (complex) Dulac germs

Autori
Peran, Dino

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
ICDEA 2022 - book of abstracts / - , 2022, 27-28

Skup
ICDEA 2022, 27th International Conference of Difference Equations and Applications

Mjesto i datum
Pariz, Francuska, 18.07.2022. - 22.07.2022

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Fixed point theory, Dulac germs, strongly hyperbolic fixed point, logarithmic transseries, normal forms

Sažetak
Dulac germs are analytic germs on an interval (0, d), d>0, that can be analytically extended to certain complex domans called standard quadratic domains, admitting certain type of asymptotic expansion called Dulac series. More precisely, Dulac series are formal series of powers multiplied by real polynomials in the formal variable log z, and they form a subset of the larger set of all logarithmic transseries (see [2]). Dulac germs appear as the first return maps of hyperbolic polycycles of analytic planar vector fields and they play an important role in the Dulac problem of nonaccumulation of limit cycles on a hyperbolic or semi-hyperbolic polycycle of an analytic planar vector field (see [3], [4]). In this talk we consider strongly hyperbolic Dulac germs f = z^α + o(z^α), α>0, α≠1. We first define the formal composition of logarithmic transseries and prove that every strongly hyperbolic logarithmic transseries f=z^α+”higher order terms”, α>0, α≠1, can be formally conjugated (normalized) to its leading term zα via parabolic conjugation g=z+”higher order terms”. On the other hand, motivated by the classical Böttcher Theorem (see [1]), we prove that every analytic complex germ f=z^α+o(z^α), α>0, α≠1, defined on certain invariant complex domain, having certain logarithmic asymptotic bound, can be analytically conjugated to the analytic germ z->z^α. In the end, we apply both results (the formal and analytic one) to prove that every strongly hyperbolic (complex) Dulac germ can be analytically conjugated (normalized) to the germ z->z^α via parabolic analytic conjugation g=z+o(z). Furthermore, the unique conjugation (normalization) g is again a (complex) Dulac germ. References [1] L. Carleson and T. W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. [2] L. van den Dries, A. Macintyre, and D. Marker, Logarithmic-exponential series, Proceedings of the International Conference “Analyse & Logique” (Mons, 1997), vol. 111, 2001, pp. 61–113. [3] Y. Il’yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, vol. 94, American Mathematical Society, Providence, RI, 1991. [4] Y. Ilyashenko and S. Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 2008.

Izvorni jezik
Engleski

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