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On the Scaling Ratios for Siegel Disks
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
2015 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 333, no 2, 931-957 p.Article in journal (Refereed) Published
Abstract [en]

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self-similar with a certain scaling ratio. The restriction of the dynamics of the quadratic polynomial to the boundary of the Siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. In this paper we provide an estimate on the quasisymmetric constant of the conjugacy, and use it to prove bounds on the scaling ratio lambda of the form alpha(gamma) <= |lambda| <= C delta(s), where s is the period of the continued fraction, and alpha is an element of (0, 1) depends on the rotation number in an explicit way, while C > 1, delta is an element of (0, 1) and gamma is an element of (0, 1) depend only on the maximum of the integers in the continued fraction expansion of the rotation number.

Place, publisher, year, edition, pages
2015. Vol. 333, no 2, 931-957 p.
National Category
Physical Sciences
URN: urn:nbn:se:uu:diva-247180DOI: 10.1007/s00220-014-2181-zISI: 000348302700011OAI: oai:DiVA.org:uu-247180DiVA: diva2:795126
Available from: 2015-03-13 Created: 2015-03-13 Last updated: 2016-02-17Bibliographically approved

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Gaidashev, Denis
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