PERIOD DOUBLING RENORMALIZATION FOR AREA-PRESERVING MAPS AND MILD COMPUTER ASSISTANCE IN CONTRACTION MAPPING PRINCIPLE

Abstract

A universal period doubling cascade analogous to the famous Feigenbaum–Coullet–Tresser period doubling has been observed in area-preserving maps of ℝ2. The existence of the "universal" map with orbits of all binary periods has been proved via a renormalization approach in [Eckmann et al., 1984] and [Gaidashev et al., 2011]. These proofs use "hard" computer assistance.In this paper, we attempt to reduce computer assistance in the argument, and present a mild computer aided proof of the analyticity and compactness of the renormalization operator in a neighborhood of a renormalization fixed point: that is, a proof that does not use generalizations of interval arithmetics to functional spaces — but rather relies on interval arithmetics on real numbers only to estimate otherwise explicit expressions. The proof relies on several instances of the Contraction Mapping Principle, which is, again, verified via mild computer assistance.