A computer-assisted proof of universality for area-preserving maps

J.-P. Eckmann; H. Koch; P. Wittwer

doi:10.1090/memo/0289

A computer-assisted proof of universality for area-preserving maps

Memoirs of the American Mathematical Society ◽

10.1090/memo/0289 ◽

1984 ◽

Vol 47 (289) ◽

pp. 0-0 ◽

Cited By ~ 15

Author(s):

J.-P. Eckmann ◽

H. Koch ◽

P. Wittwer

Keyword(s):

Computer Assisted ◽

Computer Assisted Proof ◽

Area Preserving Maps ◽

Area Preserving

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Period doubling in area-preserving maps: an associated one-dimensional problem

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000283 ◽

2010 ◽

Vol 31 (4) ◽

pp. 1193-1228 ◽

Cited By ~ 5

Author(s):

DENIS GAIDASHEV ◽

HANS KOCH

Keyword(s):

Fixed Point ◽

Period Doubling ◽

Computer Assisted ◽

Dimensional System ◽

Dimensional Problem ◽

One Dimensional ◽

Computer Assisted Proof ◽

Area Preserving Maps ◽

Image Position ◽

Area Preserving

AbstractIt has been observed that the famous Feigenbaum–Coullet–Tresser period-doubling universality has a counterpart for area-preserving maps of ℝ2. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in Eckmannet al[Existence of a fixed point of the doubling transformation for area-preserving maps of the plane.Phys. Rev. A 26(1) (1982), 720–722; A computer-assisted proof of universality for area-preserving maps.Mem. Amer. Math. Soc. 47(1984), 1–121]. As 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 Hénon-like (after a coordinate change) map:where ϕ solvesWe 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 ℂ 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 case. In this respect, the paper is an exploration of possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.

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