On the Invariant Cantor Sets of Period Doubling Type of Infinitely Renormalizable Area-Preserving Maps

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 EckmannKoch-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.

The stratification and invariant region of chaotic sea in an area-preserving Lozi map

“…E-mail address: 3014407@mail.suec.du.cn Area-preserving maps have no attractor, but have rich and very complex dynamic behavior, and can provide the simplest and most accurate way to visualize and quantify the behavior of conserved systems with two degrees of freedom [16]. Therefore, they have attracted the attention of scholars in different fields [17][18][19][20][21][22][23][24][25][26][27]. Sander et al developed a weighted Birkhoff average method to identify chaotic orbits, island chains, and rotationally invariant circles that do not depend on these symmetries [18].…”