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Boundaries of Siegel disks: Numerical studies of their dynamics and regularity

Chaos 18, 033135 (2008); doi:10.1063/1.2985856

Published 29 September 2008

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Rafael de la Llave1 and Nikola P. Petrov2
1Department of Mathematics, University of Texas, Austin, Texas 78712, USA
2Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA
Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Hölder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents. ©2008 American Institute of Physics
History: Received 16 May 2008; accepted 23 August 2008; published 29 September 2008
Permalink: http://link.aip.org/link/?CHAOEH/18/033135/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.45.-a
    Nonlinear dynamics and chaos
  • 05.10.Cc
    Renormalization group methods (statistical physics/nonlinear dynamics)
  • 02.30.Px
    Abstract harmonic analysis
  • 02.60.-x
    Numerical approximation and analysis
  • YEAR: 2008

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PUBLICATION DATA

ISSN:
1054-1500 (print)   1089-7682 (online)
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