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
Journal ArticleDOI
Rigidity of critical circle mappings I
Edson de Faria,Welington de Melo +1 more
TL;DR: In this paper, it was shown that two critical circle maps with the same rotation number in a special set are Cワン1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the Cワン0 sense.
MonographDOI
Quasiconformal surgery in holomorphic dynamics
Bodil Branner,Núria Fagella +1 more
TL;DR: This book discusses surgery and its applications in dynamical systems and actions of Kleinian groups, as well as some of the principles of surgery as applied to extensions and interpolations.
Journal ArticleDOI
The Fibonacci unimodal map
Mikhail Lyubich,John Milnor +1 more
TL;DR: In this article, the authors studied the topological, geometrical, and measure-theoretical properties of the real quadratic Fibonacci map with a degenerate critical point.
Journal ArticleDOI
Hyperbolicity of renormalization of critical circle maps
TL;DR: In this article, it was shown that the convergence of unimodal renormalization transformations to the horseshoe attractor is a geometric process, and that it is uniformly hyperbolic, with one-dimensional unstable direction.
Journal ArticleDOI
Local connectivity of some Julia sets containing a circle with an irrational rotation
TL;DR: The Julia set as discussed by the authors is the complement of the Fatou set for rational maps, and it is the set of points z c C possessing a neighbourhood on which the family of iterates {R n }n~>o is normal.
References
More filters
Book
Iterates of maps on an interval
TL;DR: Piecewise monotone functions as discussed by the authors are well-behaved piecewise monotonone functions, and the iterates of functions in S-Reductions can be reduced to homtervals.