Geometric construction and analytic representation of Julia sets of polynomial maps

Nonlinearity ◽  
2011 ◽  
Vol 24 (4) ◽  
pp. 1311-1327 ◽  
Author(s):  
O Costin ◽  
M Huang
Keyword(s):  
Julia Sets ◽  
Geometric Construction ◽  
Analytic Representation ◽  
Polynomial Maps

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.

GENERATION OF 3D JULIA SETS FROM SWITCHING POLYNOMIAL MAPS

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 205-210
Author(s):  
JIN CHENG ◽  
JIANRONG TAN ◽  
CHUNBIAO GAN
Keyword(s):  
Julia Sets ◽  
Quadratic Polynomial ◽  
Experimental Results ◽  
Polynomial Maps ◽  
Novel Approach ◽  
3D Fractals

Inspired by the study of 3D fractals based on quadratic polynomial maps,1 this paper presents a novel approach for generating 3D Julia sets by utilizing a family of switching polynomial maps in order to further enrich the form of 3D fractals. Rotational symmetries in the structures of resulting Julia sets with different switching parameters are theoretically analyzed and proved. Experimental results obtained from various parameters and coefficients in the switching maps are provided and discussed in detail. It is hoped that the investigations conducted in this paper can bring on new perspectives for the generalization of 3D fractals.

scholarly journals On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

2021 ◽  
Vol 5 (2) ◽  
pp. 55
Author(s):  
Yang Zhao ◽  
Shicun Zhao ◽  
Yi Zhang ◽  
Da Wang
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Initial Point ◽  
Visual Display ◽  
Time Algorithm ◽  
Escape Time ◽  
Polynomial Maps ◽  
Distribution Rule ◽  
Connected Area ◽  
Criterion Method

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

FIXED POINTS-JULIA SETS

2020 ◽  
Vol 9 (9) ◽  
pp. 6759-6763
Author(s):  
G. Subathra ◽  
G. Jayalalitha
Keyword(s):  
Fixed Points ◽  
Julia Sets

Dynamics of a family of orbitally continuous mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3507-3517
Author(s):  
Abhijit Pant ◽  
R.P. Pant ◽  
Kuldeep Prakash
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Real Numbers ◽  
Continuous Mappings ◽  
Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

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