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
More filters
Dissertation
Sur les applications du cercle avec un intervalle plat et flots de Cherry
TL;DR: In this paper, a description complete de the dynamique of a classe L de fonctions de degre un du cercle, supposees de classe (de two fois derivable) C^2 a l’exception de deux points ou seule la continuite is exigee, and telles qu’elles soient constantes sur un des intervalles delimite par ces derniers.
Posted Content
One-dimensional maps and Poincar\'e metric
TL;DR: In this article, the authors studied invertible compositions of one-dimensional critical circle maps with non-positive Schwarzian derivative and showed that the joint distortion of the composition is bounded.
Journal ArticleDOI
Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws
TL;DR: A family of strongly-asymmetric unimodal maps is considered that contains a Feirywasysym, and the development of the renormal map of renormalization of these maps is developed.
Posted Content
Growth sequences for circle diffeomorphisms
TL;DR: In this paper, the growth sequences of the differential for iterations of circle diffeomorphisms without periodic points were obtained. But the growth sequence was not shown for iterations with periodic points.
Posted Content
The rigidity problem for analytic critical circle maps
D. Khmelev,Michael Yampolsky +1 more
TL;DR: In this article, it was shown that if f and g are analytic critical circle mappings with the same irrational rotation number, then the conjugacy that maps the critical point of f to that of g has regularity $C^{1+\alpha}$ with a universal value of 0.
References
More filters
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.