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Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable SetsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2009 (English)In: Journal of Modern Dynamics, ISSN 1930-5311, Vol. 3, no 4, 555-587 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2009. Vol. 3, no 4, 555-587 p.
##### Keyword [en]

37E20, 37F25, 37D05, 37D20, 37C29, 37A05, 37G15, 37M99
##### National Category

Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-114002DOI: 10.3934/jmd.2009.3.555ISI: 000274246300004OAI: oai:DiVA.org:uu-114002DiVA: diva2:292548
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Available from: 2010-02-08 Created: 2010-02-08 Last updated: 2012-05-01

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 Cantor set $\cC^\infty_F$ such that the Lyapunov exponents of $F \arrowvert_{\cC^\infty_F}$ are zero, and whose Hausdorff dimension satisfies$${\rm dim}_H(\cC_F^{\infty}) < 0.5324.$$ 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.

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