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It has been observed that the famous Feigenbaum-Coullet-Tresser perioddoubling universality has a counterpart for area-preserving maps of ${\fR}^2$.A renormalization approach has been used in a "hard" computer-assisted proof ofexistence of an area-preserving map with orbits of all binary periods inEckmann et al (1984). As it is the case with all non-trivial universalityproblems in non-dissipative systems in dimensions more than one, no analyticproof of this period doubling universality exists to date.  In this paper we attempt to reduce computer assistance in the argument, andpresent a mild computer aided proof of the analyticity and compactness of therenormalization operator in a neighborhood of a renormalization fixed point:that is a proof that does not use generalizations of interval arithmetics tofunctional spaces - but rather relies on interval arithmetics on real numbersonly to estimate otherwise explicit expressions. The proof relies on severalinstance of the Contraction Mapping Principle, which is, again, verified viamild computer assistance.

Period Doubling Renormalization for Area-Preserving Maps and Mild Computer Assistance in Contraction Mapping Principle