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
Onset of stochasticity for some one-dimensional systems
TL;DR: In this article, the appearance of stochastic dynamics in two situations is studied: for one-parameter families of maps of the interval fμ, undergoing an infinite sequence of period-doubling bifurcations, and for twoparameter family of circle maps ω: x + ω + b2pisin2ππx======
Posted Content
Unbounded Regime for Circle Maps and Physical Measures for Cherry Flows
TL;DR: In this paper, the authors studied weakly order preserving circle maps with a flat interval and showed that the geometry is degenerate when the degree of the singularities is less than or equal to two.
Book ChapterDOI
Conjugations Between Two Critical Circle Maps With Non-integer Exponents
TL;DR: In this article, it was shown that if the orders of critical points are different then the map h conjugating the two homeomorphisms is a singular function, where h is the order of the critical points.
Dissertation
On C 1 -rigidity for circle maps with a break point
TL;DR: In this paper, Mazzeo et al. proved that C robust rigidity holds for circle maps with a break point satisfying a "derivatives close condition" in the fractional linear transformation (FLT) pair family.
Journal ArticleDOI
Cross-ratio inequality with infinite type of singularities
TL;DR: In this paper, a preserving orientation circle homeomorphism with infinite number of break points and finite number of singular points is considered, and the cross-ratio inequality with respect to f holds.
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.