Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets

TL;DR

This paper investigates the dynamics of infinitely renormalizable area-preserving maps near a universal fixed point, establishing the existence of a stable invariant set with zero Lyapunov exponent and bounded Hausdorff dimension, and demonstrating weak rigidity within a specific submanifold.

Contribution

It proves the existence of a stable invariant set with zero Lyapunov exponent for a class of area-preserving maps and introduces the concept of weak rigidity in a finite codimension submanifold.

Findings

Existence of a stable invariant set with zero Lyapunov exponent.

Hausdorff dimension of the stable set is at most 0.5324.

Weak rigidity of dynamics within a specific submanifold.

Abstract

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a "universal" area-preserving map $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we consider {\it infinitely renormalizable} maps -- maps on the renormalization stable manifold in some neighborhood of $F_*$ -- and study their dynamics. For all such infinitely renormalizable maps in a neighborhood of the fixed point $F_*$ we prove the existence of a "stable" invariant set $\cC^\infty_F$ such that the maximal Lyapunov exponent of $F \arrowvert_{\cC^\infty_F}$ is zero, and whose Hausdorff dimension satisfies $${\rm dim}_H(\cC_F^{\infty}) \le 0.5324.$$ We also show that there exists a submanifold, $\bW_\omega$, of finite codimension in the renormalization local stable manifold, such that for all $F\in\bW_\omega$ the set $\cC^\infty_F$ is "weakly rigid": the dynamics of any two maps in this submanifold, restricted to the stable set $\cC^\infty_F$, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.