Cylinder Renormalization of Siegel Disks

Gaidashev, Denis; Yampolsky, Michael

doi:10.1080/10586458.2007.10128991

Cited by 23 publications

(23 citation statements)

References 21 publications

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“…These scaling properties were explained in certain cases by renormalization group analysis. 2,4,11,12,36 These scaling properties suggest that the boundaries of Siegel disks can be very interesting fractal objects.…”

Section: A Some Results From Complex Dynamicsmentioning

confidence: 99%

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

Llave

Petrov

2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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 parameterization 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 parameterization 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 parameterization of the boundaries and the corresponding scaling exponents.

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Section: A Some Results From Complex Dynamicsmentioning

confidence: 99%

“…Since then this phenomenon has been a subject of extensive numerical and mathematical studies. [3][4][5][6][7][8][9][10][11][12] In this paper, we report some direct numerical calculations of the Hölder regularity of these boundaries for different rotation numbers of bounded type and for different maps.…”

Section: Introductionmentioning

confidence: 99%

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

Llave

Petrov

2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The paper [18] suggests that PACSE is evidence for the existence of a horseshoe with 1-dimensional unstable manifolds which, furthermore, satisfy some transversality conditions. Global renormalizations have been proposed for Siegel discs [23,24,25,26], unimodal maps [27,28], critical circle maps [29,30]. We hope that this paper can serve as a stimulus for the development of these theories.…”

Section: Discussionmentioning

confidence: 99%

Universal scalings of universal scaling exponents

Llave¹,

Olverá²,

Petrov³

2007

J. Phys. A: Math. Theor.

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Abstract. In the last decades, renormalization group (RG) ideas have been applied to describe universal properties of different routes to chaos (quasi-periodic, period doubling or tripling, Siegel disk boundaries, etc.). Each of the RG theories leads to universal scaling exponents which are related to the action of certain RG operators.The goal of this announcement is to show that there is a principle that organizes many of these scaling exponents. We give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations. These relations are determined by combinatorial properties of the route and become exact in an appropriate limit.

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“…The main drawback with this approach is that commuting maps do not constitute a manifold, which makes it hard to discuss some important aspects of renormalization, such as the hyperbolicity of the renormalization operator. (Similar problems are also encountered in one-dimensional dynamics, which lead to the development of alternative approaches, such as the "cylinder renormalization" for Siegel disks [125,56]. )…”

Section: Hamiltonian Systemsmentioning

confidence: 99%

Renormalization of vector fields

Koch¹

2008

Holomorphic Dynamics and Renormalization

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Abstract. These notes cover some of the recent developments in the renormalization of quasiperiodic flows. This includes skew flows over tori, Hamiltonian flows, and other flows on T d ×R . After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and invariant tori, including shearless tori, and non-differentiable tori for critical Hamiltonians. Then we discuss properties related to the spectrum of the linearized renormalization transformation, such as the accumulation rates for sequences of closed orbits. The last part describes extensions from "self-similar" to Diophantine rotation vectors. This involves sequences of renormalization transformations that are related to continued fractions expansions in one and more dimensions. Whenever appropriate, the discussion of details is restricted to special cases where inessential technical complications can be avoided.

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Cylinder Renormalization of Siegel Disks

Cited by 23 publications

References 21 publications

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

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

Universal scalings of universal scaling exponents

Renormalization of vector fields

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