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No elliptic islands for the universal area-preserving map
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics. (CAPA)
2011 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 24, no 7, 2063-2078 p.Article in journal (Refereed) Published
Abstract [en]

A renormalization approach has been used in Eckmann et al (1982) and Eckmann et al (1984) to prove the existence of a universal area-preserving map, a map with hyperbolic orbits of all binary periods. The existence of a horseshoe, with positive Hausdorff dimension, in its domain was demonstrated in Gaidashev and Johnson (2009a). In this paper the coexistence problem is studied, and a computer-aided proof is given that no elliptic islands with period less than 18 exist in the domain. It is also shown that less than 1.5% of the measure of the domain consists of elliptic islands. This is proven by showing that the measure of initial conditions that escape to infinity is at least 98.5% of the measure of the domain, and we conjecture that the escaping set has full measure. This is highly unexpected, since generically it is believed that for conservative systems hyperbolicity and ellipticity coexist.

Place, publisher, year, edition, pages
2011. Vol. 24, no 7, 2063-2078 p.
Keyword [en]
37J10, 37D05, 37E15, 37E20.
National Category
Mathematical Analysis
Research subject
URN: urn:nbn:se:uu:diva-173607DOI: 10.1088/0951-7715/24/7/008OAI: oai:DiVA.org:uu-173607DiVA: diva2:524341
Swedish Research Council, 2010-598
Available from: 2012-05-01 Created: 2012-05-01 Last updated: 2012-07-04Bibliographically approved

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