Rigidity of critical circle mappings I
Edson de Faria,Welington de Melo +1 more
TLDR
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.read more
Citations
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Self-similarity of Siegel disks and Hausdorff dimension of Julia sets
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.
MonographDOI
Quasiconformal surgery in holomorphic dynamics
Bodil Branner,Núria Fagella +1 more
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.
Journal ArticleDOI
Hyperbolicity of renormalization of critical circle maps
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.
Journal ArticleDOI
Density of hyperbolicity in dimension one
TL;DR: In this paper, the authors show that a real polynomial is hyperbolic if the real line is the union of a repelling hyperbola set and the basin of hyperbolas attracting periodic points is a basin of infinity.
References
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Journal ArticleDOI
One-Dimensional Dynamics
TL;DR: In this paper, the authors present some specific conjectures about the measure-theoretic properties of a certain class of one-dimensional quadratic functions with negative Schwarzian derivatives.
Book
Renormalization and 3-Manifolds Which Fiber over the Circle (AM-142), Volume 142
TL;DR: In this paper, a unified treatment of the construction of fixed points for renormalization and 3-dimensional hyperbolic 3-folds fibering over the circle is given.
Journal ArticleDOI
Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne
TL;DR: Gauthier-Villars as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).