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Normal forms for hyperbolic logarithmic transseries

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)

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

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