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Journal ArticleDOI

Rational rotation numbers for maps of the circle

01 Mar 1988-Communications in Mathematical Physics (Springer-Verlag)-Vol. 119, Iss: 1, pp 109-128
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.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, it was shown that two critical circle maps with the same rotation number in a special set are Cワン1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the Cワン0 sense.
Abstract: We prove that two C 3 critical circle maps with the same rotation number in a special set ? are C 1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C 0 sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ∞ critical circle maps with the same rotation number that are not C 1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.

128 citations


Cites result from "Rational rotation numbers for maps ..."

  • ...The proof of the opposite inequality is similar. ut We arrive at the following fundamental fact first proved byŚwia̧tek [18] and Herman [12]....

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  • ...All estimates performed in this paper rely heavily on thereal a-priori boundsof M. Herman [12] and G.́Swia̧tek [18]....

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  • ...We arrive at the following fundamental fact first proved by Świa̧tek [18] and Herman [12]....

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MonographDOI
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

Journal ArticleDOI
TL;DR: In this article, the authors studied the topological, geometrical, and measure-theoretical properties of the real quadratic Fibonacci map with a degenerate critical point.
Abstract: The Fibonacci recurrence of the critical orbit appeared in the work of Branner and Hubbard on complex cubic polynomials [BH, §12] and in Yoccoz's work [Yl, Y2] on quadratic ones, as the "worst" pattern of recurrence. On the other hand, a real quadratic Fibonacci map was suggested by Hofbauer and Keller [HK] as a possible candidate for a map having a "wild" attractor (that is, a set A which is the w-limit set for Lebesgue almost every orbit but is strictly smaller than the w-limit set for a generic orbit). The w-limit set of the critical orbit in [HK] possesses all known topological properties of wild attractors (compare [BL2]). In fact, we will see below that the quadratic Fibonacci map does not have a wild attractor; however, the corresponding question for a map with a degenerate critical point remains open. Actually, the first indication of the Fibonacci map appeared in the numerical work of Tsuda [T], related to the Belousova-Zhabotinskii reaction, and also in numerical work of Shibayama [Sh] (more precisely, they studied the sequence of "Fibonacci bifurcations" creating the Fibonacci map). This paper will study topological, geometrical, and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan [S]. It turns out that the situation can be understood completely and is of quite regular nature. In particular, any Fibonacci map (with negative Schwarzian and nondegenerate critical point) has an absolutely continuous invariant measure (so, we deal with a "regular" type of chaotic dynamics). It turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the Feigenbaum map: its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter. Branner and Hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits. Their "Fibonacci tableau" is a basic example, which corresponds to one particularly close and regular pattern of recurrence. If a complex quadratic map z 1-+ z2 + c realizes this Fibonacci tableau, then the

111 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the convergence of unimodal renormalization transformations to the horseshoe attractor is a geometric process, and that it is uniformly hyperbolic, with one-dimensional unstable direction.
Abstract: The renormalization theory of critical circle maps was developed in the late 1970’s–early 1980’s to explain the occurence of certain universality phenomena. These phenomena can be observed empirically in smooth families of circle homeomorphisms with one critical point, the so-called critical circle maps, and are analogous to Feigenbaum universality in the dynamics of unimodal maps. In the works of Ostlund et al. [ORSS] and Feigenbaum et al. [FKS] they were translated into hyperbolicity of a renormalization transformation. The first renormalization transformation in one-dimensional dynamics was constructed by Feigenbaum and independently by Coullet and Tresser in the setting of unimodal maps. The recent spectacular progress in the unimodal renormalization theory began with the seminal work of Sullivan [Sul1,Sul2,MvS]. He introduced methods of holomorphic dynamics and Teichmüller theory into the subject, developed a quadratic-like renormalization theory, and demonstrated that renormalizations of unimodal maps of bounded combinatorial type converge to a horseshoe attractor. Subsequently, McMullen [McM2] used a different method to prove a stronger version of this result, establishing, in particular, that renormalizations converge to the attractor at a geometric rate. And finally, Lyubich [Lyu4,Lyu5] constructed the horseshoe for unbounded combinatorial types, and showed that it is uniformly hyperbolic, with one-dimensional unstable direction, thereby bringing the unimodal theory to a completion. The renormalization theory of circle maps has developed alongside the unimodal theory. The work of Sullivan was adapted to the subject by de Faria, who constructed in [dF1,dF2] the renormalization horseshoe for critical circle maps of bounded type. Later de Faria and de Melo [dFdM2] used McMullen’s work to show that the convergence to the horseshoe is geometric. The author in [Ya1,Ya2] demonstrated the existence of the horseshoe for unbounded types, and studied the limiting situation arising when the combinatorial type of the renormalization grows without a bound. Despite the similarity in the development of the two renormalization theories up to this point, the question of hyperbolicity of the horseshoe attractor presents a notable difference. Let us recall without going into details the structure of the argument given by Lyubich in the unimodal case. The first part of Lyubich’s work was to to endow the

102 citations

Journal ArticleDOI
TL;DR: The Julia set as discussed by the authors is the complement of the Fatou set for rational maps, and it is the set of points z c C possessing a neighbourhood on which the family of iterates {R n }n~>o is normal.
Abstract: The Fatou set FR for a rational map R: C--*C is the set of points z c C possessing a neighbourhood on which the family of iterates {R n }n~>o is normal (in the sense of Montel). The Julia set JR=C--FR is the complement of the Fatou set. (The monographs [CG], [Be], [St] provide introductions to the theory of iteration of rational maps.) Let 0E ]0, 1 [ Q be an irrational number and write it as a continued fraction

100 citations

References
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Book
01 Jan 1938
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
Abstract: This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford, Cambridge, Aberdeen, and other universities. It is neither a systematic treatise on the theory of numbers nor a 'popular' book for non-mathematical readers. It contains short accounts of the elements of many different sides of the theory, not usually combined in a single volume; and, although it is written for mathematicians, the range of mathematical knowledge presupposed is not greater than that of an intelligent first-year student. In this edition, the main changes are in the notes at the end of each chapter. Sir Edward Wright seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present, both in the notes and in the text, a reasonably accurate account of the present state of knowledge.

5,972 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos and the complementary set to the stability interval is a Cantor set of fractal dimension $D=0.87.
Abstract: It is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos. The complementary set to the stability intervals is a Cantor set of fractal dimension $D=0.87$. This exponent is found to be universal for a large class of functions.

238 citations

Book ChapterDOI
01 Jan 1977

120 citations

Journal ArticleDOI
TL;DR: On considere la classe #7B-A des applications C ∞ f:[0, 1]→[0,1] telles que f(0)=f(1)=0 et f a unique point critique C ∈(0, 2) as mentioned in this paper.
Abstract: On considere la classe #7B-A des applications C ∞ f:[0,1]→[0,1] telles que f(0)=f(1)=0 et f a un unique point critique C∈(0,1). Si le point critique de f∈#7B-A est non plat alors f n'a pas d'intervalle errant

97 citations


"Rational rotation numbers for maps ..." refers methods in this paper

  • ...It is known that ifh has no flat critical points then it is possible to D estimate the for h and, which is more interesting, for large order iterates of h. The estimations of that kind are used in [ 4 ] and [7] and prove to be useful there....

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