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

### Abstract

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

Received 16 May 2008
Accepted 23 August 2008
Published online 29 September 2008

Lead Paragraph:
According to a celebrated theorem by Siegel, under certain arithmetic conditions, the dynamical behavior of the iterates of a holomorphic map around a neutrally stable fixed point of the map is very simple—the iterates of the map fill densely analytic topological circles around the critical point. In the domain where the iterates exhibit such behavior (called Siegel disks), there exists a complex analytic change of variables that makes the map locally a multiplication by a complex number of modulus 1. On the boundary of a Siegel disk, however, the dynamical behavior of the iterates of the map is dramatically different—for example, the boundary is not a smooth curve. The iterates on the boundary of the Siegel disk exhibit scaling properties that have motivated the development of a renormalization group description. The dynamically natural parametrization of the boundary of the Siegel disk has low regularity. We compute accurately the natural parametrization of the boundary and apply methods from harmonic analysis to compute the Hölder exponents of the parametrizations of the boundaries for different maps with different rotation numbers and with different orders of criticality of their critical points.

Acknowledgments:
The work of both authors has been partially supported by NSF grants. We thank Amit Apte, Lukas Geyer, and Arturo Olvera for useful discussions. This work would not have been possible without access to excellent publicly available software for compiler, graphics, extended precision, especially GMP (Ref. 43). We thank the Mathematics Department at University of Texas at Austin for the use of computer facilities.

Article outline:

I. INTRODUCTION
II. SIEGEL DISKS AND THEIR BOUNDARIES
A. Some results from complex dynamics
B. Parametrization of the boundary of a Siegel disk
C. Continued fraction expansions and rational approximations
D. Scaling exponents
III. HÖLDER REGULARITY AND SCALING PROPERTIES OF THE BOUNDARIES OF SIEGEL DISKS—NUMERICAL METHODS AND RESULTS
A. Some results from harmonic analysis
B. Algorithms used
C. Visual explorations
D. Numerical values of the Hölder regularity
E. Importance of the phases of Fourier coefficients
F. Data on the scaling exponents
IV. CALCULATION OF THE SIEGEL RADIUS AND THE AREA OF THE SIEGEL DISK
A. Calculation of Siegel radius
B. Calculation of the area of the Siegel disk
V. AN UPPER BOUND OF THE REGULARITY OF THE CONJUGACY
VI. CONCLUSIONS

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2008-09-29

2016-02-14

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