A note on the Julia set of a rational function

1995 ◽  
Vol 118 (3) ◽  
pp. 477-485 ◽  
Author(s):  
S. D. Letherman ◽  
R. M. W. Wood
Keyword(s):  
Rational Function ◽  
Complex Dynamics ◽  
Julia Set ◽  
Julia Sets ◽  
Inverse Image ◽  
Iteration Algorithm ◽  
Finite Sets ◽  
The Subject

The purpose of this note is to present a few facts about the Julia set of a rational function that are well known to the experts in the subject of complex dynamics but whose documented exposition in the literature seems to need a little clarification. For example, under conditions set out in Theorem 1, the Julia set of a rational function can be expressed as the limit of a sequence of finite sets. In particular, for certain choices of a point α, the Julia set is the limit as n increases of the inverse image sets R−n(α). This formulation is widely exploited in the backwards iteration algorithm to produce computer illustrations of Julia sets (see for example Section 5·4 of [14]).

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

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.

DIMENSIONS OF JULIA SETS OF HYPERBOLIC MEROMORPHIC FUNCTIONS

2001 ◽  
Vol 33 (6) ◽  
pp. 689-694 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Meromorphic Function ◽  
Box Dimensions

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

scholarly journals Julia Set with Trigonometric Function

2019 ◽  
Vol 8 (2S7) ◽  
pp. 115-119
Keyword(s):  
Complex Number ◽  
Complex Dynamics ◽  
Julia Set ◽  
Julia Sets ◽  
New Class ◽  
Iteration Function ◽  
Cubic Polynomials ◽  
Constant Complex ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Julia sets are generated by initializing a complex number z = x + yi where z is then iterated using the iteration function fc (z)= zn 2 + c, where n indicates the number of iteration and c is a constant complex number. Recently, study of cubic Julia sets was introduced in Noor Orbit (NO) with improved escape criterions for cubic polynomials. In this paper, we investigate the complex dynamics of different functions and apply the iteration function to generate an entire new class of Julia sets. Here, we introduce different types of orbits on cubic Julia sets with trigonometric functions. The two functions to investigate from Julia sets are sine and cosine functions.

Hausdorff dimension of Julia sets of complex Hénon mappings

1996 ◽  
Vol 16 (4) ◽  
pp. 849-861 ◽  
Author(s):  
A. Verjovsky ◽  
H. Wu
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Invariant Sets ◽  
Analytic Dependence ◽  
Fractal Sets ◽  
Complex Dimension ◽  
Real Analytic

AbstractThe Hausdorff dimension of closed invariant sets under diffeomorphisms is an interesting concept as it is a measure of their complexity. The theory of holomorphic dynamical systems provides us with many examples of fractal sets and, in particular, a theorem of Ruelle [Ru1] shows that the Hausdorff dimension of the Julia set depends real analytically onfiffis a rational function of ℂ and the Julia setJoffis hyperbolic. In this paper we generalize Ruelle's result for complex dimension two and show the real analytic dependence of the Hausdorff dimension of the corresponding Julia sets of hyperbolic Hénon mappings.

Julia sets of rational functions are uniformly perfect

1993 ◽  
Vol 113 (3) ◽  
pp. 543-559 ◽  
Author(s):  
A. Hinkkanen
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Uniformly Perfect

AbstractLetfbe a rational function of degree at least two. We shall prove that the Julia setJ(f) offis uniformly perfect. This means that there is a constantc∈(0, 1) depending onfonly such that wheneverz∈J(f) and 0 <r< diamJ(f) thenJ(f) intersects the annulus.

scholarly journals Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

2021 ◽  
pp. 1-18
Author(s):  
YÛSUKE OKUYAMA
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Projective Line ◽  
Closed Field ◽  
Fatou Component ◽  
Algebraically Closed Field ◽  
Absolute Value ◽  
Uniformly Perfect ◽  
Uniform Perfectness

Abstract We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

Fractal analysis and control of the competition model

2016 ◽  
Vol 09 (03) ◽  
pp. 1650045 ◽  
Author(s):  
Mianmian Zhang ◽  
Yongping Zhang
Keyword(s):  
Feedback Control ◽  
Mathematical Models ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Julia Set ◽  
Julia Sets ◽  
Competition Model ◽  
Simulation Results ◽  
Population Competition ◽  
And Control

Lotka–Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods.

The evolution of the public sphere

Kybernetes ◽  
2019 ◽  
Vol 49 (9) ◽  
pp. 2201-2219
Author(s):  
José J. Blanco
Keyword(s):  
Public Sphere ◽  
Design Methodology ◽  
Complex Dynamics ◽  
Media Theory ◽  
Language Games ◽  
Content Type ◽  
The Public ◽  
The Subject ◽  
The Public Sphere ◽  
Dominant Paradigm

Purpose The purpose of this study is to rethink the issue of publicity from a cross-cultural and evolutionary perspective. Design/methodology/approach Assuming that there is a dominant paradigm in the studies of the public sphere centered on Habermas’ ideas, media theory (and especially Luhmann who is considered as a media theorist) is selected as a new context that provides different concepts, ideas, language games and metaphors that allow the re-foundation of the study of publicity. Findings Publicity as a social structure emerges – and acquires different forms during history – out of the complex dynamics resulting from the interaction between success media, such as power, and different kinds of dissemination media. Originality/value A research into the forms of publicity not only promotes awareness of the ubiquity of the phenomenon across cultural evolution, but also offers tools to make new discoveries and systematize what is already known about the subject and its ramifications.

scholarly journals Wiman–Valiron Discs and the Dimension of Julia Sets

2020 ◽  
Author(s):  
James Waterman
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Julia Sets ◽  
Singular Values ◽  
Simply Connected ◽  
Set Of Points ◽  
Meromorphic Map

Abstract We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

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