Journal of Modern Dynamics (JMD)

Dynamics of the universal area-preserving map associated with period-doubling: Stable sets

Pages: 555 - 587, Issue 4, October 2009      doi:10.3934/jmd.2009.3.555

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Denis Gaidashev - Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden (email)
Tomas Johnson - Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden (email)

Abstract: It is known that the famous Feigenbaum-Coullet-Tresser period-doubling universality has a counterpart for area-preserving maps of $R^2$. A renormalization approach was used in [11] and [12] in a computer-assisted proof of the existence of a "universal'' area-preserving map $F_*$, that is, a map with orbits of all binary periods $2^k, k \in N$. In this paper, we consider infinitely renormalizable maps, which are 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 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 bi-Lipschitz transformation, which preserves the Hausdorff dimension.

Keywords:  Area-preserving maps, rigidity, period-doubling, renormalization, rigorous computations.
Mathematics Subject Classification:  37E20, 37F25, 37D05, 37D20, 37C29, 37A05, 37G15, 37M99.

Received: May 2009;      Revised: November 2009;      Available Online: January 2010.