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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Posted Content
Scalings in circle maps III
TL;DR: In this article, the Hausdorff dimension of the non-wandering set of non-differentiable circle maps with a flat spot has been studied and a sharp transition from degenerate geometry to bounded geometry has been found.
Proceedings ArticleDOI
Circle homeomorphism with infinite type of singularities
TL;DR: In this article, it was shown that the cross-ratio distortion with respect to a given circle homeomorphism with infinite number of break and finite number of singular points is bounded.
Posted Content
On the Hausdorff dimension of invariant measures for multicritical circle maps
TL;DR: In this paper, the Hausdorff dimension of the unique invariant measure of multicritical circle maps without periodic points was shown to be Ω(C^3 )-approximation.
Book ChapterDOI
Renormalization, Zygmund Smoothness and the Epstein Class
TL;DR: The folding permutation as mentioned in this paper is a permutation of order n if irreducible, and it may be accomplished by a continuous mapping f of the real line to itself which folds the line once.
Book ChapterDOI
Dynamics on the Circle
TL;DR: In this paper, the authors review several results on the dynamics of circle maps, including the theory of circle diffeomorphism where the combinatorial aspects goes back to Poincare followed by the topological description of the dynamics by Denjoy and the geometric aspects by Herman and Yoccoz.
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.