Abstract
Since its inception in the 1970s at the hands of Feigenbaum and, independently, Coullet and Tresser the study of renormalization operators in dynamics has been very successful at explaining universality phenomena observed in certain families of dynamical systems. The first proof of existence of a hyperbolic fixed point for renormalization of area-preserving maps was given by Eckmann et al. (Mem Am Math Soc 47(289):vi+122, 1984). However, there are still many things that are unknown in this setting, in particular regarding the invariant Cantor sets of infinitely renormalizable maps. In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the Eckmann–Koch–Wittwer renormalization fixed point is always contained in a Lipschitz curve but never contained in a smooth curve. This extends previous results by de Carvalho, Lyubich and Martens about strongly dissipative maps of the plane close to unimodal maps to the area-preserving setting. The method used for constructing the Lipschitz curve is very similar to the method used in the dissipative case but proving the nonexistence of smooth curves requires new techniques.
Highlights
The study of renormalization techniques in dynamics began in the 1970s in independent efforts by Feigenbaum [6,7] and Coullet and Tresser [20] to explain the observed universality phenomena in families of maps on the interval undergoing period doubling bifurcation
It has been shown that the infinitely renormalizable maps, i.e., those contained in the stable manifold of the renormalization fixed point, have invariant Cantor sets
The dynamics of any two infinitely renormalizable maps restricted to their respective invariant Cantor sets are topologically conjugate
Summary
The study of renormalization techniques in dynamics began in the 1970s in independent efforts by Feigenbaum [6,7] and Coullet and Tresser [20] to explain the observed universality phenomena in families of maps on the interval undergoing period doubling bifurcation. For this operator they show, in addition to the previous results, that the invariant Cantor sets are not rigid They show that a topological invariant of infinitely renormalizable strongly dissipative Hénon-like maps called the average Jacobian is an obstruction to rigidity. In Eckmann et al [5] introduced a renormalization operator for area-preserving maps of period doubling type of the plane and proved, using computer assistance, the existence of a hyperbolic fixed point This explains previously observed universality phenomena in families of such maps. Further investigations of this renormalization have been done by Gaidashev and Johnson [8,9,10] and by Gaidashev et al [11] In these papers they prove existence of period doubling invariant Cantor sets for all infinitely renormalizable maps and show that they are rigid.
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