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Normal forms for transseries and Dulac germs

Transseries are formal (possibly infinite) sums of monomials that are formal products of iterated exponentials, powers and iterated logarithms, with real coefficients (see e.g. [6], [21]). We consider here a subclass of logarithmic transseries which contain only powers and iterated logarithms. Transseries appear in many problems in mathematics ( [3], [11]) and physics ( [1]) as asymptotic expansions of certain important maps. In dynamics, for example, transseries are related to the famous Dulac’s problem ( [7]) of non- accumulation of limit cycles on a hyperbolic or semi-hyperbolic polycycle of a planar analytic vector field. The problem was solved independently by Ilyashenko ( [10], [11], [12]) and Écalle ( [3]), but the proofs are so far not well- understood, at least in the semi-hyperbolic case. The study of the accumulation of limit cycles on a polycycle is naturally related to the study of fixed points of the first return map of a polycycle (see e.g. [32]). The first return map of a hyperbolic polycycle is an analytic map on interval (0 ; d), d > 0, which has a transserial asymptotic expansion at zero. In particular, its asymptotic expansion at zero is a logarithmic series involving only polynomials in logarithms attached to each power, which is called a Dulac series (see e.g. [12], [32]). The proof of the Dulac problem strongly relies on the existence of a logarithmic asymptotic expansion of the first return map. Although Dulac gave the proof ( [7]) of the mentioned problem, there was an imprecision in his proof. In particular, at some point in the proof, the statement that every first return map of a hyperbolic polycycle is uniquely determined by its asymptotic expansion is used. This is not correct in general for non-analytic maps on the real line, due to the possibility of adding exponentially small terms, as opposed to the case of analytic maps and their Taylor expansions. Ilyashenko corrected this imprecision in [11] by proving that every such map can be analytically extended to a sufficiently large complex domain called a standard quadratic domain and by applying the Phragmen-Lindelöf Theorem (a maximum modulus principle on an unbounded complex domain, see e.g. [11], [12]). The existence of such analytic extension makes possible to conclude that the first return map is uniquely determined by its logarithmic asymptotic expansion. In this dissertation, we consider the so-called Dulac germs (called almost regular germs in [12]), i.e., analytic germs on (0 ; d), d > 0, that have a Dulac series as their asymptotic expansion at zero, and that can be analytically extended to a standard quadratic domain. On the one hand, we consider normal forms of logarithmic transseries (the formal part), and, on the other hand, analytic normalizations of hyperbolic and strongly hyperbolic Dulac germs (the analytic part). We also generalize to their complex counterparts, called hyperbolic and strongly hyperbolic complex Dulac germs. We prove as well that, for hyperbolic and strongly hyperbolic Dulac germs, the formal normalizations are asymptotic expansions of their analytic normalizations. In proving that the formal transserial normalization is an asymptotic expansion of the analytic normalization, in general, there is a problem of a choice of the summation rule at the limit ordinal steps. In particular, given some map f on open interval (0 ; d), d > 0, we want to assign to the map f its asymptotic expansion at zero in power-iterated logarithm scale. Up to the first limit ordinal it can be done following the usual Poincaré algorithm, contrary to the limit ordinal steps where we have multiple choices of intermediate sums. Therefore, we have to determine a summation rule at limit ordinal steps, which vary from problem to problem (see e.g. integral summation rule in [20], [22]). Luckily, for hyperbolic and strongly hyperbolic Dulac germs the formal normalizations are Dulac series, so standard Poincaré algorithm suffices. On the other hand, in case of parabolic Dulac germs, it is proved in [22] that, in general, the formal normalization is of order type strictly bigger than w. Normal forms and normalizations of standard power series are already known (see e.g. [4], [12], [16]). Furthermore, in previous papers ( [21], [22]), normal forms for logarithmic transseries were obtained only for power-logarithm transseries, i.e., logarithmic transseries that contain only powers and the first iterate of logarithm. The techniques used in [21] are based on a transfinite algorithm of successive changes of variables. Here, we generalize these results to a larger class of logarithmic transseries containing also iterated logarithms. Additionally, as a normalization process we use fixed point theorems on various complete metric spaces of logarithmic transseries. In this way, normalizations are given as limits (in appropriate topologies) of Picard sequences. This is important for the future work because we think that this approach is better for revealing the summation rule at limit ordinal steps. In proving the existence of the analytic normalization of a hyperbolic Dulac germ, we generalize the classical Koenigs Theorem (see e.g. [4], [14], [24]) for linearization of analytic hyperbolic diffeomorphisms at zero. Recently, there have been some improvements of this result for various classes of maps not necessarily analytic at 0. One such generalization is a result of Dewsnap and Fischer [5] for C1 real maps on an open interval around zero with power-logarithmic asymptotic bounds. In this dissertation, we prove a linearization theorem for analytic maps with power-logarithmic asymptotic bounds on invariant complex domains, that can be seen as a generalization of both Koenigs Theorem and the result of Dewsnap and Fischer from [5, Theorem 2.2]. In particular, we apply the mentioned linearization theorem to obtain the analytic linearization of hyperbolic (complex) Dulac germs. Finally, we also generalize the Bottcher Theorem ¨ (see e.g. [4], [24]) for germs of strongly hyperbolic analytic diffeomorphisms at zero to strongly hyperbolic complex Dulac germs on standard quadratic domains.

logarithmic transseries, order of transseries, normal forms, normalization, linearization, formal and analytic classification, (complex) Dulac germs, Dulac series, standard quadratic domains, local fixed point theory, fixed point theorems, iteration theory, Koenigs sequence

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