On holomorphic critical quasi circle maps
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The so-called Herman-Światek theorem was generalized to holomorphic self-homeomorphisms of quasi-circles in this paper, which implies an unpublished theorem of Michel Herman: if a Siegel disk or Arnold-Herman ring for a rational map has a boundary component, which is a quasi-circle containing a critical point, then the associated rotation number is Diophantine of exponent 2.Abstract:
The so-called Herman–Światek theorem is generalized to holomorphic self-homeomorphisms of quasi-circles. This result implies an unpublished theorem of Michel Herman: If a Siegel disk or Arnold–Herman ring for a rational map has a boundary component, which is a quasi-circle containing a critical point, then the associated rotation number is Diophantine of exponent 2.read more
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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.
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Polynomial Siegel disks are typically Jordan domains
TL;DR: For typical rotation numbers, this article proved that polynomial Siegel disks are Jordan domains with boundaries containing at least one critical point, where the critical point is a critical point.
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Some transcendental entire functions with irrationally indifferent fixed points
Masashi Kisaka,Hiroto Naba +1 more
TL;DR: In this article , a polynomial-like mappings with irrationally indifferent fixed points were constructed for the set of all transcendental entire functions of the form P(z) \exp (Q(z)) where $P$ and $Q$ are polynomials.
A priori bounds and degeneration of Herman rings with bounded type rotation number
TL;DR: In this article , it was shown that the boundaries of Herman rings of bounded type and of the simplest configuration are quasicircles with dilatation depending only on the degree and the rotation number.
References
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Journal ArticleDOI
Rational rotation numbers for maps of the circle
TL;DR: 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.
Journal ArticleDOI
On critical circle homeomorphisms
TL;DR: In this article, it was shown that an analytic circle homeomorphism without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if an only if its rotation number is of constant type.