Naslov
Normal forms for hyperbolic logarithmic transseries

Autori
Peran, Dino ; Resman, Maja ; Rolin, Jean-Philippe ; Servi, Tamara

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

Izvornik
Book of abstracts, NoLineal 20-21 / - , 2021, 55-55

Skup
12th International Conference on Nonlinear Mathematics and Physics, NoLineal 20-21 Online

Mjesto i datum
Madrid, Španjolska, 30.06.2021. - 02.07.2021

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
logarithmic transseries, linearization, formal classification, iteration theory, Koenigs sequence

Sažetak
Generally speaking, transseries are formal sums of iterated exponentials, powers and iterated logarithms (see e.g. [3]). We are focused here on the so-called hyperbolic logarithmic transseries f which do not contain exponentials and have λz, 0 < λ < 1, as their first term, i.e. f = λz + ... . Such logarithmic series naturally appear in considering dynamics of analytic planar vector fields, for example, as asymptotic expansions of first return maps of hyperbolic or semi-hyperbolic polycycles. We recall here the famous Dulac problem of non-accumulation of limit cycles to hyperbolic and semi-hyperbolic polycycles, solved independently by Ilyashenko [2] and Ecalle. ´ Normal forms, from aspect of dynamics, represent simplest objects (i.e. germs, series, vector fields) that are, in some sense (formal, C∞, analytic, . . . ), equivalent to the original object. We obtain formal normal forms and formal normalizations for hyperbolic logarithmic transseries, using Banach fixed point theorem. This result represents a generalization of normalization results obtained in [4]. The techniques used for normalization in [4] are different, in the sense that they are based on transfinitely many term-wise elementary changes of variables. We use our result to prove a generalization of the standard Koenigs’ linearization theorem (see e.g. [1]) to hyperbolic logarithmic transseries and to hyperbolic germs on complex domains admitting logarithmic asymptotic expansions. [1] L. Carleson and T. W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag (New York, 1993) [2] Y. Ilyashenko and S. Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society (Providence, RI, 2008) [3] L. van den Dries, A. Macintyre, and D. Marker, Logarithmic-exponential series, Proceedings of the International Conference ”Analyse & Logique” (Mons, 1997), vol. 111, (2001) 61-113 [4] P. Mardeˇsi´c, M. Resman, J.-P. Rolin, and V. Zupanovi´c, ˇ Normal forms and embeddings for power-log transseries, Adv. Math. 303, 888-953 (2016)

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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