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Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: Journal of Modern Dynamics, ISSN 1930-5311, Vol. 3, no 4, 555-587 p.Article in journal (Refereed) Published
Abstract [en]

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.

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
Available from: 2010-02-08 Created: 2010-02-08 Last updated: 2012-05-01

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Gaidashev, DenisJohnson, Tomas

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