Journal ArticleDOI
Rational rotation numbers for maps of the circle
TLDR
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.read more
Citations
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Peer Review
Dennis Sullivan's Work on Dynamics
TL;DR: In this paper , an overview of the major contributions of Sullivan's major contributions to the area of Dynamical Systems is provided. But the readers are not provided with a comprehensive overview of these contributions.
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Mating the Basilica with a Siegel Disk
TL;DR: In this article, a quadratic polynomial with a fixed Siegel disc of bounded type was shown to be conformally mateable with the basilica polynomials.
Journal ArticleDOI
Renormalization of Analytic Multicritical Circle Maps with Bounded Type Rotation Numbers
Automorphic measures and invariant distributions for circle dynamics
Edson de Faria,Pablo Guarino +1 more
TL;DR: In this article , it was shown that the space of invariant distributions of order 1 of any multicritical circle map is one-dimensional, spanned by the unique invariant measure.
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On the scaling ratios for Siegel disks
TL;DR: In this article, the authors provided an estimate on the quasisymmetric constant of the conjugacy, and used it to prove bounds on the scaling ratio of the Siegel disk of a quadratic polynomial.
References
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Book
An Introduction to the Theory of Numbers
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.
Journal ArticleDOI
Complete Devil's Staircase, Fractal Dimension, and Universality of Mode- Locking Structure in the Circle Map
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.
Journal ArticleDOI
A structure theorem in one dimensional dynamics
W. de Melo,S. van Strien +1 more
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.