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Dynamics of the universal areapreserving map associated with perioddoubling: Stable sets
1.  Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden, Sweden 
For all such infinitely renormalizable maps in a neighborhood of the fixed point $F_*$, we prove the existence of a "stable'' invariant Cantor set $\l^\infty_F$ such that the Lyapunov exponents of $F _{\l^\infty_F}$ are zero and whose Hausdorff dimension satisfies
$\text{dim}_H(\l_F^{\infty}) < 0.5324.$
We also show that there exists a submanifold, $W_\omega$, of finite codimension in the renormalization local stable manifold such that for all $F\in W_\omega$, the set $\l^\infty_F$ is "weakly rigid'': the dynamics of any two maps in this submanifold, restricted to the stable set $\l^\infty_F$, are conjugate by a biLipschitz transformation, which preserves the Hausdorff dimension.
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