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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Journal ArticleDOI
Nombre de rotation presque sûr des endomorphismes du cercle affines par morceaux
TL;DR: In this article, the authors define le nombre de rotation presque sur de certains endomorphismes du cercle de degre un, i.e., the number of rotations of a point at each point in a given point.
Journal ArticleDOI
On cross-ratio distortion and Schwarz derivative
TL;DR: In this article, asymptotic estimates for the cross-ratio distortion with respect to a smooth or holomorphic function in terms of its Schwarz derivative are derived. But they do not consider the case where the function is smooth.
Posted Content
Quasisymmetric orbit-flexibility of multicritical circle maps
Edson de Faria,Pablo Guarino +1 more
TL;DR: In this paper, it was shown that the number of equivalence classes is uncountable for critical circle maps with Diophantine rotation number in the full Lebesgue measure set.
Journal ArticleDOI
Herman's Theory Revisited
TL;DR: In this paper, it was shown that a smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class (D_\delta) is smoothly conjugate to a rigid rotation.
Journal ArticleDOI
Renormalizations of circle hoemomorphisms with a single break point
TL;DR: In this paper, it was shown that for circle diffeomorphisms with irrational rotation number 1, a linear fractional function can be approximated in the Euclidean space in the C √ 1 + L √ 2 -norm.
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.