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A renormalization approach to invariant circles in area-preserving maps. (English) Zbl 1194.37068
Summary: Kadanoff and Shenker introduced a renormalisation approach to invariant circles in area-preserving maps. This paper makes more precise the connection between invariant circles and the renormalisation operator. Restricting attention to noble rotation numbers, the stability of a simple fixed point of the renormalisation is analysed, corresponding to a linear twist map. It is found to be essentially attracting, so that noble circles persist under perturbation, giving a new view on KAM theory. Shenker and Kadanoff found evidence for another fixed point, corresponding to a map with a non-smooth noble circle. Further evidence is given in this paper. It has essentially only one unstable direction, and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle. Finally, noble circles are shown to be locally most robust, in an important sense.

37E10 Dynamical systems involving maps of the circle
37E40 Dynamical aspects of twist maps
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