2008/11/16 by Denis Gaidashev, Gaidashev, Denis, Hans Koch +1
Mathematics · #28D05 #37E20 #37E30 #37F25 #65P30 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS #msc:28D05 #msc:37E20 #msc:37E30 #msc:37F25 #msc:65P30
paper · pdf · doi:10.48550/arxiv.0811.2588
Typos removed. An argument about sequential compactness replaced by one based on Tikhonov-Schauder theorem
arxiv created 2009/06/04 · arxiv updated 2009/12/01
It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of \fieldR2. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving 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 non-trivial universality problems in non-dissipative 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 area-preserving maps is almost one dimensional, in the sense that it is close to the following Henon-like map: H^*(x,u)=(ϕ(x)-u,x-ϕ(ϕ(x)-u)), where ϕ solves ϕ(x)=2 \over λ ϕ(ϕ(λx))-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 ϕ together with verification of several inequalities and inclusions of subsets of \fieldC numerically. Finally, we suggest an analytic approach to the full period doubling problem for area-preserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.