Dynamics of the universal area-preserving map associated with period-doubling: Stable sets
Denis Gaidashev - 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 $\text{dim}_H(\l_F^{\infty}) < 0.5324.$ 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.
Received: May 2009; Revised: November 2009; Available Online: January 2010. |