 08216 Denis Gaidashev, Hans Koch
 Period doubling in areapreserving maps: an associated onedimensional problem
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Nov 16, 08

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Abstract. It has been observed that the famous FeigenbaumCoulletTresser period doubling universality has a counterpart for areapreserving maps of $\field{R}^2$. A renormalization approach has been used in a computerassisted proof of existence of an areapreserving map with orbits of all binary periods by J.P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all nontrivial universality problems in nondissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date.
We argue that the period doubling renormalization fixed point for areapreserving maps is almost one dimensional, in the sense that it is close to the following Henonlike map: $$H^*(x,u)=(\phi(x)u,x\phi(\phi(x)u )),$$ where $\phi$ solves $$\phi(x)={2 \over \lambda} \phi(\phi(\lambda x))x.$$
We then give a ``proof'' of existence of solutions of small analytic perturbations of this one dimensional problem, and describe some of the properties of this solution.
The ``proof'' consists of an analytic argument for factorized inverse branches of $\phi$ together with verification of several inequalities and inclusions of subsets of $\field{C}$ numerically.
Finally, we suggest an analytic approach to the full period doubling problem for areapreserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a nontrivial renormalization problem in a conservative twodimensional system.
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