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Description
We consider germs $f$ with asymptotic logarithmic bounds, i.e. $f(z)=\lambda z+o(zL(z))$, $0<|\lambda |<1$, uniformly as $|z|\to 0$, on Riemann surface of the logarithm, where $L(z)$ is, roughly speaking, some product of powers of iterated logarithms . Instead of the standard $z$-chart, we consider the logarithmic chart $\zeta :=-\log z$ which is a global chart for Riemann surface of the logarithm. In this chart germ $f$ can be written in the form $\widehat{f}(\zeta )=\zeta -\log \lambda + o(\zeta^{-1}\widehat{L}(\zeta ))$, uniformly as $\mathrm{Re} \, \zeta \to + \infty $. In order to linearize such germs, we are motivated by famous Koenigs' Theorem about linearization of analytic hyperbolic diffeomorphisms at zero. But instead of open balls around zero, we consider open domains on Riemann surface of the logarithm which spiral around the origin. The spiraling bound of the domain can be of any kind, but we choose special domains which we call the admissible ones. The importance of these domains is their $\widehat{f}$-invariance. Therefore, we consider the Koenigs' sequence $(\widehat{f}^{\circ n})$ and prove its uniform convergence towards the unique linearization of the germ $\widehat{f}$ on some admissible domain. This result is simultaneously a generalization of Koenigs' Theorem and the recent Dewsnapp-Fischer's result about the linearization of real germs with logarithmic asymptotic bounds. We apply our result to the problem of linearization of Dulac germs which are specal kinds of such germs that appear naturally in solutions of the famous Dulac problem of nonaccumulation of limit cycles on a hyperbolic polycycle of an analytic planar vector field. This is joint work with M. Resman, J.P. Rolin and T. Servi.
