scholarly journals GEOMETRIC LIMITS OF MANDELBROT AND JULIA SETS UNDER DEGREE GROWTH

2012 ◽  
Vol 22 (12) ◽  
pp. 1250301 ◽  
Author(s):  
SUZANNE HRUSKA BOYD ◽  
MICHAEL J. SCHULZ
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Julia Sets ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Geometric Limit ◽  
The Family ◽  
Mandelbrot Sets

First, for the family Pn,c(z) = zn + c, we show that the geometric limit of the Mandelbrot sets Mn(P) as n → ∞ exists and is the closed unit disk, and that the geometric limit of the Julia sets J(Pn,c) as n tends to infinity is the unit circle, at least when |c| ≠ 1. Then, we establish similar results for some generalizations of this family; namely, the maps z ↦ zt + c for real t ≥ 2 and the rational maps z ↦ zn + c + a/zn.

LIMITING JULIA SETS FOR SINGULARLY PERTURBED RATIONAL MAPS

2008 ◽  
Vol 18 (10) ◽  
pp. 3175-3181 ◽  
Author(s):  
MARK MORABITO ◽  
ROBERT L. DEVANEY
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Julia Set ◽  
Singularly Perturbed ◽  
Parameter Plane ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Open Set ◽  
The Family

In this paper, we consider the family of rational maps given by [Formula: see text] where n ≥ 2, and λ is a complex parameter. When λ = 0 the Julia set is the unit circle, as is well known. But as soon as λ is nonzero, the Julia set explodes. We show that, as λ tends to the origin along n - 1 special rays in the parameter plane, the Julia set of Fλ converges to the closed unit disk. This is somewhat unexpected, since it is also known that, if a Julia set contains an open set, it must be the entire Riemann sphere.

scholarly journals CHECKERBOARD JULIA SETS FOR RATIONAL MAPS

2013 ◽  
Vol 23 (02) ◽  
pp. 1330004 ◽  
Author(s):  
PAUL BLANCHARD ◽  
FİGEN ÇİLİNGİR ◽  
DANIEL CUZZOCREO ◽  
ROBERT L. DEVANEY ◽  
DANIEL M. LOOK ◽  
...  
Keyword(s):  
Rotation Number ◽  
Julia Sets ◽  
Conjugacy Classes ◽  
Rational Maps ◽  
Root Of Unity ◽  
The Family ◽  
Mandelbrot Sets ◽  
The Given

In this paper, we consider the family of rational maps [Formula: see text] where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or [Formula: see text] where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

scholarly journals Geometric Limits of Julia Sets of Maps zn + exp(2πiθ) as n → ∞

2015 ◽  
Vol 25 (08) ◽  
pp. 1530021 ◽  
Author(s):  
Scott R. Kaschner ◽  
Reaper Romero ◽  
David Simmons
Keyword(s):  
Unit Circle ◽  
Julia Sets ◽  
Geometric Limit

We show that the geometric limit as n → ∞ of the Julia sets J(Pn,c) for the maps Pn,c(z) = zn + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle.

scholarly journals Dirichlet integral and Picard principle

1982 ◽  
Vol 86 ◽  
pp. 85-99
Author(s):  
Mitsuru Nakai ◽  
Toshimasa Tada
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Ideal Boundary ◽  
Dirichlet Integral ◽  
Hölder Continuous ◽  
Relative Boundary ◽  
The Family ◽  
The Ideal

A density P on the punctured unit disk Ω:0 < |z| <1 is a 2-form P(z)dxdy whose coefficient P(z) is a real valued nonnegative locally Hölder continuous function on the closed punctured unit disk Ω:0< |z| <≦1. Here we consider Ω as an end of the punctured sphere 0 < |z| ≦ + + so that the point z = 0 is viewed as the ideal boundary δΣ of Σ and the unit circle |z| = 1 as the relative boundary δΣ of Σ. We denote by D = D(Σ) the family of densities on Σ.

scholarly journals Lacunary Möbius Fractals on the Unit Disk

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 91
Author(s):  
L. K. Mork ◽  
Keith Sullivan ◽  
Darin J. Ulness
Keyword(s):  
Automorphism Group ◽  
Unit Disk ◽  
Parameter Space ◽  
Rotational Symmetry ◽  
Julia Sets ◽  
Three Dimensional ◽  
Open Unit ◽  
Open Unit Disk ◽  
Dimensional Parameter ◽  
Mandelbrot Sets

Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk D, which is a subgroup of the Möbius transformations. The behavior, dimension, dynamics, and sensitivity of filled-in Julia sets and Mandelbrot sets to variables will be discussed in detail. Additionally, several visualizations of this three-dimensional parameter space will be presented.

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 Inner composition of analytic mappings on the unit disk

1991 ◽  
Vol 14 (2) ◽  
pp. 221-226 ◽  
Author(s):  
John Gill
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Analytic Functions ◽  
Continued Fraction ◽  
Unique Fixed Point ◽  
Iteration Theory ◽  
Basic Theorem ◽  
Closed Unit Disk ◽  
Compositional Structure ◽  
Compositional Structures

A basic theorem of iteration theory (Henrici [6]) states thatfanalytic on the interior of the closed unit diskDand continuous onDwithInt(D)f(D)carries any pointz ϵ Dto the unique fixed pointα ϵ Doff. That is to say,fn(z)→αasn→∞. In [3] and [5] the author generalized this result in the following way: LetFn(z):=f1∘…∘fn(z). Thenfn→funiformly onDimpliesFn(z)λ, a constant, for allz ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structuresf1∘…∘fn(z)where thefj's are analytic onInt(D)and continuous onDwithInt(D)fj(D), but essentially random. Applications include analytic functions defined by this process.

scholarly journals CONNECTIVITY OF JULIA SETS FOR SINGULARLY PERTURBED RATIONAL MAPS

2013 ◽  
pp. 239-245 ◽  
Author(s):  
ROBERT L. DEVANEY ◽  
ELIZABETH D. RUSSELL
Keyword(s):  
Julia Sets ◽  
Singularly Perturbed ◽  
Rational Maps

scholarly journals Strong Differential Subordination Results for Multivalent Analytic Functions Associated with Dziok-Srivastava Operator

2019 ◽  
pp. 159-168
Author(s):  
Abbas Kareem Wanas ◽  
Hala Abbas Mehdi
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Unit Disk ◽  
Strong Differential Subordination

In this paper, by making use of the principle of strong subordination, we establish some interesting properties of multivalent analytic functions defined in the open unit disk and closed unit disk of the complex plane associated with Dziok-Srivastava operator.

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