On critical circle homeomorphisms
01 Sep 1998-Boletim Da Sociedade Brasileira De Matematica (Springer-Verlag)-Vol. 29, Iss: 2, pp 329-351
TL;DR: In this article, it was shown 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.
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.
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TL;DR: In this paper, it was shown that the Hausdorff dimension of the Julia set J(f) is strictly less than two, and if θ is a quadratic irrational (such as the golden mean), then the Siegel disk is self-similar about the critical point.
Abstract: Let f(z) = ez+ z, where θ is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: • the Hausdorff dimension of the Julia set J(f) is strictly less than two; and • if θ is a quadratic irrational (such as the golden mean), then the Siegel disk for f is self-similar about the critical point. In the latter case, we also show the rescaled first-return maps converge exponentially fast to a system of commuting branched coverings of the complex plane.
138 citations
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.
122 citations
TL;DR: In this article, it was shown that the Julia set of the quadratic polynomial P has a Siegel disk whose boundary is a Jordan curve passing through the critical point of P.
Abstract: Let 0 << 1 be an irrational number with continued fraction expan- sion =( a1 ;a 2 ;a 3 ;::: ), and consider the quadratic polynomial P : z 7! e 2i z +z 2 . By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if log an = O( p n )a sn !1 ; then the Julia set of P is locally-connected and has Lebesgue measure zero. In particular, it follows that for almost every 0 << 1, the quadratic P has a Siegel disk whose boundary is a Jordan curve passing through the critical point of P.B y standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.
94 citations
TL;DR: In this paper, the existence of quadratic polynomials having a Julia set with positive Lebesgue measure was proved and the ideas of the proof and the techniques involved were presented.
Abstract: We recently proved the existence of quadratic polynomials having a Julia set with positive Lebesgue measure. We present the ideas of the proof and the techniques involved.
72 citations
Cites background from "On critical circle homeomorphisms"
...If α ∈ S1, the polynomial Pα has a Siegel disk bounded by a quasicircle containing the critical point (see [D1], [He], [ Sw ])....
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TL;DR: In this article, the authors studied the one-dimensional parameter space of cubic polynomials in the complex plane which have a fixed Siegel disk of rotation number θ, where θ is a given irrational number of Brjuno type.
Abstract: We study the one-dimensional parameter space of cubic polynomials in the complex plane which have a fixed Siegel disk of rotation number θ, where θ is a given irrational number of Brjuno type. The main result of this work is that when θ is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also show that these boundaries vary continuously as one moves in the parameter space. This is most nontrivial near the set of cubics with both critical points on the boundary of their Siegel disk. We prove that this locus is a Jordan curve in the parameter space. Most of the techniques and results can be generalized to polynomials of higher degrees.
59 citations
References
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Book•
01 Jan 1980
TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.
Abstract: Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails. The past twenty years have been an amazingly fruitful period of research in algorithmic graph theory and structured families of graphs. Especially important have been the theory and applications of new intersection graph models such as generalizations of permutation graphs and interval graphs. These have lead to new families of perfect graphs and many algorithmic results. These are surveyed in the new Epilogue chapter in this second edition. New edition of the "Classic" book on the topic Wonderful introduction to a rich research area Leading author in the field of algorithmic graph theory Beautifully written for the new mathematician or computer scientist Comprehensive treatment
4,090 citations
Book•
01 Jan 1973
TL;DR: In this article, the authors define a geometric definition of a quasiconformal mapping and apply it to the Hilbert transformation to a set of dimensions of a circle domain and a ring domain.
Abstract: I. Geometric Definition of a Quasiconformal Mapping.- to Chapter I.- 1. Topological Properties of Plane Sets.- 2. Conformal Mappings of Plane Domains.- 3. Definition of a Quasiconformal Mapping.- 4. Conformal Module and Extremal Length.- 5. Two Basic Properties of Quasiconformal Mappings.- 6. Module of a Ring Domain.- 7. Characterization of Quasiconformality with the Help of Ring Domains.- 8. Extension Theorems for Quasiconformal Mappings.- 9. Local Characterization of Quasiconformality.- II. Distortion Theorems for Quasiconformal Mappings.- to Chapter II.- 1. Ring Domains with Extremal Module.- 2. Module of Grotzsch's Extremal Domain.- 3. Distortion under a Bounded Quasiconformal Mapping of a Disc.- 4. Order of Continuity of Quasiconformal Mappings.- 5. Convergence Theorems for Quasiconformal Mappings.- 6. Boundary Values of a Quasiconformal Mapping.- 7. Quasisymmetric Functions.- 8. Quasiconformal Continuation.- 9. Circular Dilatation.- III. Auxiliary Results from Real Analysis.- to Chapter III.- 1. Measure and Integral.- 2. Absolute Continuity.- 3. Differentiability of Mappings of Plane Domains.- 4. Module of a Family of Arcs or Curves.- 5. Approximation of Measurable Functions.- 6. Functions with Lp-derivatives.- 7. Hubert Transformation.- IV. Analytic Characterization of a Quasiconformal Mapping.- to Chapter IV.- 1. Analytic Properties of a Quasiconformal Mapping.- 2. Analytic Definition of Quasiconformality.- 3. Variants of the Geometric Definition.- 4. Characterization of Quasiconformality with the Help of the Circular Dilatation.- 5. Complex Dilatationl.- V. Quasiconformal Mappings with Prescribed Complex Dilatation.- to Chapter V.- l. Existence Theorem.- 2. Local Dilatation Measures.- 3. Removable Point Sets.- 4. Approximation of a Quasiconformal Mapping.- 5. Application of the Hilbert Transformation to Quasiconformal Mappings21l.- 6. Conformality at a Point.- 7. Regularity of a Mapping with Prescribed Complex Dilatation.- VI. Quasiconformal Functions.- to Chapter VI.- 1. Geometric Characterization of a Quasiconformal Function.- 2. Analytic Characterization of a Quasiconformal Function.
1,359 citations
TL;DR: In this article, the authors consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, and prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0.
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.
140 citations
TL;DR: In this paper, it was shown that if a rational function f of the Riemann sphere of degree 2 leaves invariant a singular domain C (a disk or a ring) on which the rotation number off satisfies a diophantine condition, provided that each boundary component of C contains critical point off, then each boundary part of C is injective.
Abstract: Generalizing a result of E. Ghys, we prove a general theorem that implies that if a rational functionf of the Riemann sphere of degree ≧2 leaves invariant a singular domainC (a disk or a ring) on which the rotation number off satisfies a diophantine condition, provided that on
$$\bar C$$
f is injective, then each boundary component ofC contains critical point off. The injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off(z)=z
n
+a, withneℕ,n≧2 andaeℂ. We also show that the singular disks, associated to periodic elliptic points off(z)=e
az
that satisfy a diophantine condition, are unbounded in ℂ. In the end of the paper, we give a survey of the theory of iteration of entire functions of ℂ.
107 citations
"On critical circle homeomorphisms" refers methods in this paper
...type. His solution was based on quasiconformal surgery, see [ 4 ], and a theorem about the quasi-symmetric conjugacy between an analytic...
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TL;DR: The classical criterion for a circle diffeomorphism to be topologically and topologically conjugate to an irrational rigid rotation was given by Denjoy [1] as discussed by the authors.
Abstract: The classical criterion for a circle diffeomorphism to be topologically
conjugate to an irrational rigid rotation was given by Denjoy [1]
In [5] one of us gave a new criterion There is an
example satisfying Denjoy's bounded variation condition rather than
the Zygmund condition of [5], and vice versa This paper will give
the third criterion which is implied by either of the above criteria
37 citations