scholarly journals A family of rational maps with buried Julia components

2014 ◽  
Vol 35 (6) ◽  
pp. 1846-1879 ◽  
Author(s):  
SÉBASTIEN GODILLON
Keyword(s):  
Explicit Formula ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Jordan Curves ◽  
Polynomial Map

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However, all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least five. This paper introduces a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover, this family contains some rational maps of degree three with explicit formula that answers a question McMullen raised.

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.

CONTROL AND SYNCHRONIZATION OF JULIA SETS OF THE COMPLEX PERTURBED RATIONAL MAPS

2013 ◽  
Vol 23 (05) ◽  
pp. 1350083 ◽  
Author(s):  
YONGPING ZHANG
Keyword(s):  
Optimal Control ◽  
Function Method ◽  
Control Method ◽  
Control Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Control And Synchronization ◽  
Function Synchronization ◽  
Optimal Control Function

The dynamical and fractal behaviors of the complex perturbed rational maps [Formula: see text] are discussed in this paper. And the optimal control function method is taken on the Julia set of this system. In this control method, infinity is regarded as a fixed point to be controlled. By substituting the driving item for an item in the optimal control function, synchronization of Julia sets of two such different systems is also studied.

Variation of Hausdorff dimension of Julia sets

1993 ◽  
Vol 13 (1) ◽  
pp. 167-174 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Hausdorff Dimension ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Analytic Family ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

AbstractLet (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfieswhere ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

scholarly journals Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

2010 ◽  
Vol 30 (6) ◽  
pp. 1869-1902 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Jordan Curve ◽  
Riemann Sphere ◽  
Julia Set ◽  
Julia Sets ◽  
John Domain ◽  
Unbounded Component ◽  
Random Dynamics ◽  
Jordan Curves ◽  
Polynomial Maps

AbstractWe investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere $\CCI $) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that, for almost every sequence $\gamma \in \Gamma ^{\NN }$, the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of $\CCI {\setminus } J_{\gamma }$ is a John domain, and the bounded component of $\CC {\setminus } J_{\gamma }$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

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.

scholarly journals Invariant Jordan curves of Sierpiński carpet rational maps

2016 ◽  
Vol 38 (2) ◽  
pp. 583-600 ◽  
Author(s):  
YAN GAO ◽  
PETER HAÏSSINSKY ◽  
DANIEL MEYER ◽  
JINSONG ZENG
Keyword(s):  
Jordan Curve ◽  
Julia Set ◽  
Rational Maps ◽  
Rational Map ◽  
Sierpinski Carpet ◽  
Jordan Curves ◽  
Postcritically Finite ◽  
Sierpiński Carpet

In this paper, we prove that if $R:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer $n_{0}$, such that, for any $n\geq n_{0}$, there exists an $R^{n}$-invariant Jordan curve $\unicode[STIX]{x1D6E4}$ containing the postcritical set of $R$.

scholarly journals Rational maps whose Julia sets are Cantor circles

2013 ◽  
Vol 35 (2) ◽  
pp. 499-529 ◽  
Author(s):  
WEIYUAN QIU ◽  
FEI YANG ◽  
YONGCHENG YIN
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Cantor Set ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

scholarly journals On (non-)local connectivity of some Julia sets

2014 ◽  
Author(s):  
Alexandre Dezotti ◽  
Pascale Roesch
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Complete Characterization ◽  
Rational Maps ◽  
Local Connectivity ◽  
Rational Map ◽  
Critical Set ◽  
Quadratic Polynomials ◽  
Non Local

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally connected. The question of the local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. This chapter thus proposes some conjectures and develops a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials. This model presents the structure of what the post-critical set in that setting should be.

Julia sets converging to filled quadratic Julia sets

2012 ◽  
Vol 34 (1) ◽  
pp. 171-184 ◽  
Author(s):  
ROBERT T. KOZMA ◽  
ROBERT L. DEVANEY
Keyword(s):  
Singular Perturbations ◽  
Julia Set ◽  
Julia Sets ◽  
Quadratic Polynomial ◽  
Mandelbrot Set ◽  
Rational Maps ◽  
Hausdorff Topology ◽  
Quadratic Map ◽  
Hyperbolic Component

AbstractIn this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

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.

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