USING THE FATOU SET TO STUDY THE JULIA SET

2018 ◽  
pp. 174-218
Keyword(s):  
Julia Set ◽  
Fatou Set

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

scholarly journals Any counterexample to Makienko’s conjecture is an indecomposable continuum

2009 ◽  
Vol 29 (3) ◽  
pp. 875-883 ◽  
Author(s):  
CLINTON P. CURRY ◽  
JOHN C. MAYER ◽  
JONATHAN MEDDAUGH ◽  
JAMES T. ROGERS Jr
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Fatou Set ◽  
Indecomposable Continuum

AbstractMakienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

scholarly journals Orbifold expansion and entire functions with bounded Fatou components

2021 ◽  
pp. 1-40
Author(s):  
LETICIA PARDO-SIMÓN
Keyword(s):  
General Class ◽  
Entire Functions ◽  
Julia Set ◽  
Holomorphic Dynamics ◽  
Fatou Set ◽  
Local Connectivity ◽  
Hyperbolic Functions ◽  
Hyperbolic Orbifold ◽  
Transcendental Entire Functions ◽  
Fatou Components

Abstract Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

scholarly journals Structurally Similar Sets of Transcendental Entire Function

2018 ◽  
Vol 4 ◽  
pp. 11-16
Author(s):  
Bishnu Hari Subedi
Keyword(s):  
Entire Function ◽  
Complex Plane ◽  
Full Text ◽  
Complex Dynamics ◽  
Julia Set ◽  
Transcendental Entire Function ◽  
Fatou Set ◽  
Classical Sense

In complex dynamics, the complex plane is partitioned into invariant subsets. In classical sense, these subsets are of course Fatou set and Julia set. Rest of the abstract available with the full text

SINGULAR PERTURBATIONS OF zn WITH MULTIPLE POLES

2008 ◽  
Vol 18 (04) ◽  
pp. 1085-1100 ◽  
Author(s):  
SEBASTIAN M. MAROTTA
Keyword(s):  
Singular Perturbations ◽  
Julia Set ◽  
Complex Parameter ◽  
Fatou Set ◽  
Small Complex ◽  
The Family ◽  
Topological Characteristics ◽  
Multiple Poles

We study the dynamics of the family of complex maps given by fλ(z) = zn + λ/((z - a)da(z - b)db) where n ≥ 2 is an integer and λ is an arbitrarily small complex parameter. We focus on the topological characteristics of the Julia set and the Fatou set of fλ(z). We prove that despite the large amount of possibilities there are only four different cases that correspond to different positions and orders of the poles a and b.

Ergodic properties of semi-hyperbolic functions with polynomial Schwarzian derivative

2010 ◽  
Vol 53 (2) ◽  
pp. 471-502
Author(s):  
Volker Mayer ◽  
Mariusz Urbański
Keyword(s):  
Schwarzian Derivative ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Hyperbolic Functions ◽  
Absolutely Continuous ◽  
Ergodic Properties ◽  
Asymptotic Values ◽  
Conformal Measure ◽  
Finite Invariant Measure

AbstractThe ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative are investigated under the condition that the function is semi-hyperbolic, i.e. the asymptotic values of the Fatou set are in attracting components and the asymptotic values in the Julia set are boundedly non-recurrent. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure m with minimal exponent h; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a σ-finite invariant measure μ absolutely continuous with respect to m. Our main result states that μ is finite if and only if the order ρ of the function f satisfies the condition h > 3ρ/(ρ+1). When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's Formula, showing that the exponent h equals the Hausdorff dimension of the Julia set of f.

Random iterations of polynomials of the form $z^2+c_n$: connectedness of Julia sets

1999 ◽  
Vol 19 (5) ◽  
pp. 1221-1231 ◽  
Author(s):  
RAINER BRÜCK ◽  
MATTHIAS BÜGER ◽  
STEFAN REITZ
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Fatou Set ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Necessary And Sufficient

For a sequence $(c_n)$ of complex numbers we consider the quadratic polynomials $f_{c_n}(z):=z^2+c_n$ and the sequence $(F_n)$ of iterates $F_n:= f_{c_n} \circ \dotsb \circ f_{c_1}$. The Fatou set $\mathcal{F}_{(c_n)}$ is by definition the set of all $z \in \widehat{\mathbb{C}}$ such that $(F_n)$ is normal in some neighbourhood of $z$, while the complement of $\mathcal{F}_{(c_n)}$ is called the Julia set $\mathcal{J}_{(c_n)}$. The aim of this paper is to study the connectedness of the Julia set $\mathcal{J}_{(c_n)}$ provided that the sequence $(c_n)$ is bounded and randomly chosen. For example, we prove a necessary and sufficient condition for the connectedness of $\mathcal{J}_{(c_n)}$ which implies that $\mathcal{J}_{(c_n)}$ is connected if $|c_n| \le \frac{1}{4}$, while it is almost surely disconnected if $|c_n| \le \delta$ for some $\delta>\frac{1}{4}$.

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