scholarly journals Subhyperbolic rational maps on boundaries of hyperbolic components

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yan Gao ◽  
Luxian Yang ◽  
Jinsong Zeng
Keyword(s):  
Critical Points ◽  
Rational Maps ◽  
Rational Map ◽  
Hyperbolic Component ◽  
Quasiconformal Deformation ◽  
Geometrically Finite

<p style='text-indent:20px;'>In this paper, we prove that every quasiconformal deformation of a subhyperbolic rational map on the boundary of a hyperbolic component <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{H} $\end{document}</tex-math></inline-formula> still lies on <inline-formula><tex-math id="M2">\begin{document}$ \partial \mathcal{H} $\end{document}</tex-math></inline-formula>. As an application, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components.</p>

scholarly journals Jarník and Julia; a Diophantine analysis for parabolic rational maps for Geometrically Finite Kleinian Groups with Parabolic Elements

2002 ◽  
Vol 91 (1) ◽  
pp. 27 ◽  
Author(s):  
B. O. Stratmann ◽  
M. Urbański
Keyword(s):  
Number Theory ◽  
Multifractal Analysis ◽  
Julia Sets ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Rational Map ◽  
Geometrically Finite ◽  
Conformal Measure ◽  
Further Development

In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarník in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a "weak multifractal analysis" of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan's famous dictionary translating between the theory of Kleinian groups and the theory of rational maps.

Limiting behavior of Julia sets of singularly perturbed rational maps

2014 ◽  
Author(s):  
Robert L. Devaney
Keyword(s):  
Singular Perturbations ◽  
Julia Sets ◽  
Singularly Perturbed ◽  
Rational Maps ◽  
Rational Map ◽  
Limiting Behavior ◽  
Jordan Curves ◽  
Hyperbolic Component ◽  
Dynamical Properties ◽  
Sierpinski Gaskets

This chapter surveys dynamical properties of the families fsubscript c,𝜆(z) = zⁿ + c + λ‎/zᵈ for n ≥ 2, d ≥ 1, with c corresponding to the center of a hyperbolic component of the Multibrot set. These rational maps produce a variety of interesting Julia sets, including Sierpinski carpets and Sierpinski gaskets, as well as laminations by Jordan curves. The chapter describes a curious “implosion” of the Julia sets as a polynomial psubscript c = zⁿ + c is perturbed to a rational map fsubscript c,𝜆. In this way the chapter shows yet another way of producing rational maps through “singular” perturbations of complex polynomials.

Rational maps whose Fatou components are Jordan domains

1996 ◽  
Vol 16 (6) ◽  
pp. 1323-1343 ◽  
Author(s):  
Kevin M. Pilgrim
Keyword(s):  
Jordan Curve ◽  
Critical Points ◽  
Julia Set ◽  
Rational Maps ◽  
Rational Map ◽  
Fatou Component ◽  
Jordan Curves ◽  
Fatou Components

AbstractWe prove: If f(z) is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of f is a Jordan curve. If f(z) is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps f with a Fatou component ω satisfying f(ω) = ω and f|∂ω not topologically conjugate to the dynamics of any polynomial on its Julia set.

scholarly journals THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS

2002 ◽  
Vol 85 (2) ◽  
pp. 467-492 ◽  
Author(s):  
TIM BEDFORD ◽  
ALBERT M. FISHER ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Julia Set ◽  
Density Theorem ◽  
Finite Union ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Rational Map ◽  
Open Set ◽  
Almost All ◽  
Scenery Flow ◽  
Measure Of Maximal Entropy

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.

Density of hyperbolicity for rational maps with Cantor Julia sets

2011 ◽  
Vol 32 (5) ◽  
pp. 1711-1726 ◽  
Author(s):  
WENJUAN PENG ◽  
YONGCHENG YIN ◽  
YU ZHAI
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

Meromorphic multifunctions in complex dynamics

1992 ◽  
Vol 12 (1) ◽  
pp. 39-52 ◽  
Author(s):  
L. Baribeau ◽  
T. J. Ransford
Keyword(s):  
Critical Points ◽  
Complex Dynamics ◽  
Julia Set ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Analytic Family ◽  
Limit Sets ◽  
Upper Semicontinuous ◽  
Open Set ◽  
Similar Vein

AbstractLet {RA} be an analytic family of rational maps and denote by j(λ) the Julia set of Rλ. We prove that the upper semicontinuous regularization j(λ) of j(λ) (which coincides with j(λ) for all λ in a dense open set) is a meromorphic multifunction, and give applications that illustrate the instability of Julia sets. In a similar vein, we also consider forward orbits of critical points and limit sets of Kleinian groups.

scholarly journals CONSTRUCTING HERMAN RINGS BY TWISTING ANNULUS HOMEOMORPHISMS

2009 ◽  
Vol 86 (1) ◽  
pp. 139-143
Author(s):  
XIUMEI WANG ◽  
GAOFEI ZHANG
Keyword(s):  
Rotation Number ◽  
Critical Points ◽  
Rational Map ◽  
Herman Ring

AbstractLet F(z) be a rational map with degree at least three. Suppose that there exists an annulus $H \subset \widehat {\mathbb {C}}$ such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.

scholarly journals RATIONAL MAPS ADMITTING MEROMORPHIC INVARIANT LINE FIELDS

2009 ◽  
Vol 80 (3) ◽  
pp. 454-461 ◽  
Author(s):  
XIAOGUANG WANG
Keyword(s):  
Chebyshev Polynomial ◽  
Rational Maps ◽  
Invariant Line ◽  
Rational Map ◽  
Line Field

AbstractIt is shown that a rational map of degree at least 2 admits a meromorphic invariant line field if and only if it is conformally conjugate to either an integral Lattès map, a power map, or a Chebyshev polynomial.

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