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On the Invariant Cantor Sets of Period Doubling Type of Infinitely Renormalizable Area-Preserving Maps
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
2018 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 358, no 3, p. 1027-1039Article in journal (Refereed) Published
Abstract [en]

Since its inception in the 1970s at the hands of Feigenbaum and, independently, Coullet and Tresser the study of renormalization operators in dynamics has been very successful at explaining universality phenomena observed in certain families of dynamical systems. The first proof of existence of a hyperbolic fixed point for renormalization of area-preserving maps was given by Eckmann et al. (Mem Am Math Soc 47(289):vi+122, 1984). However, there are still many things that are unknown in this setting, in particular regarding the invariant Cantor sets of infinitely renormalizable maps. In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point is always contained in a Lipschitz curve but never contained in a smooth curve. This extends previous results by de Carvalho, Lyubich and Martens about strongly dissipative maps of the plane close to unimodal maps to the area-preserving setting. The method used for constructing the Lipschitz curve is very similar to the method used in the dissipative case but proving the nonexistence of smooth curves requires new techniques.

Place, publisher, year, edition, pages
SPRINGER , 2018. Vol. 358, no 3, p. 1027-1039
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-354521DOI: 10.1007/s00220-017-3018-3ISI: 000428046300006OAI: oai:DiVA.org:uu-354521DiVA, id: diva2:1233510
Funder
The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), 2012-2153Available from: 2018-07-18 Created: 2018-07-18 Last updated: 2018-07-18Bibliographically approved

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