Existence of a fixed point of the doubling transformation for area-preserving maps of the plane

1982 ◽  
Vol 26 (1) ◽  
pp. 720-722 ◽  
Author(s):  
J. -P. Eckmann ◽  
H. Koch ◽  
P. Wittwer
Keyword(s):  
Fixed Point ◽  
Area Preserving Maps ◽  
Area Preserving

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

2010 ◽  
Vol 31 (4) ◽  
pp. 1193-1228 ◽  
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.

Unique resonant normal forms for area-preserving maps at an elliptic fixed point

Nonlinearity ◽  
2009 ◽  
Vol 22 (4) ◽  
pp. 783-810 ◽  
Author(s):  
Vassili Gelfreich ◽  
Natalia Gelfreikh
Keyword(s):  
Fixed Point ◽  
Normal Forms ◽  
Area Preserving Maps ◽  
Area Preserving

scholarly journals A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds

1987 ◽  
Vol 7 (3) ◽  
pp. 463-479 ◽  
Author(s):  
Stephan Pelikan ◽  
Edward E. Slaminka
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Fixed Point Index ◽  
Compact Sets ◽  
Point Index ◽  
Area Preserving Maps ◽  
Area Preserving

AbstractThe study of area preserving maps of manifolds has an extensive history in the theory of dynamical systems. One interest has been in the behaviour of such maps near an isolated fixed point. In 1974 Carl Simon proved the existence of an upper bound for the index of an isolated fixed point for Ck area preserving diffeomorphisms of a surface. We extend his result to homeomorphisms of an orientable two manifold. The proof utilizes the notion of free modification, developed by Morton Brown, and enlarges the scope of the problem to the consideration of ‘nice’ measures, i.e. uniformly equivalent to Lebesgue measure on compact sets. By suitably modifying the homeomorphism and the measure, we obtain the following theorem.

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