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Journal ArticleDOI

On holomorphic critical quasi circle maps

01 Oct 2004-Ergodic Theory and Dynamical Systems (Cambridge University Press)-Vol. 24, Iss: 5, pp 1739-1751

Abstract: The so-called Herman–Światek theorem is generalized to holomorphic self-homeomorphisms of quasi-circles. This result implies an unpublished theorem of Michel Herman: If a Siegel disk or Arnold–Herman ring for a rational map has a boundary component, which is a quasi-circle containing a critical point, then the associated rotation number is Diophantine of exponent 2.
Topics: Holomorphic function (60%), Diophantine equation (58%), Critical point (mathematics) (55%), Rotation number (53%), Exponent (52%)
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MonographDOI
Bodil Branner1, Núria Fagella2Institutions (2)
01 Jan 2013
TL;DR: This book discusses surgery and its applications in dynamical systems and actions of Kleinian groups, as well as some of the principles of surgery as applied to extensions and interpolations.
Abstract: Preface Introduction 1. Quasiconformal geometry 2. Extensions and interpolations 3. Preliminaries on dynamical systems and actions of Kleinian groups 4. Introduction to surgery and first occurrences 5. General principles of surgery 6. Soft surgeries with a contribution by X. Buff and C. Henriksen 7. Cut and paste surgeries with contributions by K. M. Pilgrim, Tan Lei and S. Bullett 8. Cut and paste surgeries with sectors with a contribution by A. L. Epstein and M. Yampolsky 9. Trans-quasiconformal surgery with contributions by C. L. Petersen and P. Haissinsky Bibliography Symbol index Index.

100 citations


Posted Content
Abstract: We prove that for typical rotation numbers polynomial Siegel disks are Jordan domains with boundaries containing at least one critical point.

7 citations


Cites background from "On holomorphic critical quasi circl..."

  • ...The formulation of Lemma 4.17 is strongly inspired by [18] where a similar area estimate was established by using Petersen’s puzzle construction for the Douady-Ghys Blaschke model....

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  • ...I would like to express my deep thanks to Prof. Carsten Lunde Petersen who spent many hours discussing with me on an early version of the manuscript during his visit of Nanjing in March, 2012....

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  • ...This is the essential challenge in generalizing Petersen-Zakeri’s theorem to polynomial maps of higher degrees....

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  • ...In 2002 Petersen and Zakeri proved that for typical rotation numbers, a quadratic Siegel disk is a Jordan domain with a critical point on its boundary....

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  • ...These critical points may interact with each other if one wants to adapt Petersen’s puzzle construction to the Blaschke model B....

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Journal ArticleDOI
Abstract: We study the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplicity more than two but also an asymptotic value. Another one is a critical value with free dynamics under iterations. We prove that if the multiplicity of the superattracting fixed point is large enough, then the restriction of the transcendental entire function near the Siegel point is a quadratic-like map. Therefore the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality. As its applications, the logarithmic lift of the above transcendental entire function has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

References
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Journal ArticleDOI
Abstract: We consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, that is maps like $$x \to x + t + \frac{c}{{2\pi }}\sin (2\pi x)(\bmod 1)$$ withc=1. We prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0. In other words, the intervals on which frequency-locking occurs fill up the set of full measure.

132 citations


Journal ArticleDOI
Grzegorz Światek1Institutions (1)
Abstract: We prove that an analytic circle homeomorphism without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if an only if its rotation number is of constant type. Next, we consider automorphisms of quasi-conformal Jordan curves, without periodic orbits and holomorphic in a neighborhood. We prove a “Denjoy theorem” that such maps are conjugated to a rotation on the circle.

43 citations


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