engleski

Linearization of hyperbolic logarithmic transseries and Dulac germs

Logarithmic transseries are formal sums of powers and iterated logarithms with real coefficients. We consider hyperbolic logarithmic transseries f = λz + ..., 0 < λ < 1. In dynamics, transseries are associated with the Dulac’s problem of non-accumulation of limit cycles on a hyperbolic or semi-hyperbolic polycycle of an analytic planar vector field, which is solved independently by Ilyashenko and Ecalle. Every real analytic germ ´ on h0, di, d > 0, with Dulac series as its asymptotic expansion, which can be expanded on some complex domain called standard quadratic domain, is called Dulac germ. We obtain normal forms of hyperbolic transseries, which are, roughly speaking, the simplest transseries which are conjugated to the original one. In fact, we generalize results from [1], but using different techniques. In particular, we obtain normalizations using Banach fixed point theorem. By Koenigs’ theorem, we know that complex analytic diffeomorphism f(z) = λz+o(z), 0 < |λ| < 1, can be linearized. We find necessary and sufficient condition for hyperbolic transseries to be linearized and we apply these results to prove linearization theorem for hyperbolic Dulac germs on standard quadratic domains, which can be seen as a generalization of the mentioned Koenigs’ theorem. References [1] P. Mardeˇsi´c, M. Resman, J.-P. Rolin, and V. Zupanovi´c, ˇ Normal forms and embeddings for power-log transseries, Adv. Math. 303 (2016), 888–953.

logarithmic transseries, linearization, Dulac germs, Dulac series, standard quadratic domains, iteration theory, Koenigs sequence

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