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

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\text{fR}}^2$. 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 $F_{\ast }$ – a map with orbits of all binary periods $2^k,k\in \text{fN}$. In this paper, we consider {\it infinitely renormalizable} maps – maps on the renormalization stable manifold in some neighborhood of $F_{\ast }$ – and study their dynamics. For all such infinitely renormalizable maps in a neighborhood of the fixed point $F_{\ast }$ we prove the existence of a “stable” invariant set ${\text{cC}}_F^{\mathrm{\infty }}$ such that the maximal Lyapunov exponent of $F{\text{arrowvert}}_{{\text{cC}}_F^{\mathrm{\infty }}}$ is zero, and whose Hausdorff dimension satisfies $${\mathrm{d}\mathrm{i}\mathrm{m}}_H({\text{cC}}_F^{\mathrm{\infty }})\le \mathrm{0.5324.}$$ We also show that there exists a submanifold, ${\text{bW}}_{\omega }$, of finite codimension in the renormalization local stable manifold, such that for all $F\in {\text{bW}}_{\omega }$ the set ${\text{cC}}_F^{\mathrm{\infty }}$ is “weakly rigid”: the dynamics of any two maps in this submanifold, restricted to the stable set ${\text{cC}}_F^{\mathrm{\infty }}$, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.