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Abstract:  It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of the plane. A renormalization approach to the problem has been suggested in the 80-s by J.-P. Eckmann et al. As it is the case with all non-trivial universality problems in conservative systems in dimension more than one, to date there is no analytic proof of existence of an infinite cascade of period-doubling bifurcations.

We argue that the "universal" map exhibiting period doubling is almost one-dimensional in an appropriate sense, and present an "almost" analytic proof of existence of the renormalization fixed point for this one-dimensional problem. We also suggest an approach to the original problem based on its proximity to the one-dimesional.