Geometric limits of Julia  sets and connectedness locus of the family of polynomials Pc(z) = zn + czk

2019 ◽  
Vol 34 (4) ◽  
pp. 668-684
Author(s):  
Alexandre Miranda Alves
Keyword(s):  
Julia Sets ◽  
The Family ◽  
Connectedness Locus

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.

BIFURCATION CURVES OF A FAMILY OF ONE-DIMENSIONAL BIQUADRATIC POLYNOMIAL MAPS, DEPENDING ON THE COMPLEX VARIABLE, AND TWO REAL PARAMETERS

2010 ◽  
Vol 20 (12) ◽  
pp. 4119-4125
Author(s):  
HISASHI ISHIDA ◽  
TSUYOSHI ITOH
Keyword(s):  
Fixed Points ◽  
Complex Variable ◽  
Elementary Proof ◽  
Precise Description ◽  
One Dimensional ◽  
Polynomial Maps ◽  
The Family ◽  
Connectedness Locus

Sun and Yin [2007] had presented a precise description of the connectedness locus of the family of real biquadratic polynomials {pa,b(z) = (z2 + a)2 + b}. We shall first give an elementary proof of their result. Second, we shall give a precise description of the sets of parameters (a, b) such that the family {pa,b} has attracting fixed points.

scholarly journals Mating quadratic maps with the modular group II

2019 ◽  
Vol 220 (1) ◽  
pp. 185-210
Author(s):  
Shaun Bullett ◽  
Luna Lomonaco
Keyword(s):  
Modular Group ◽  
Parameter Family ◽  
Quadratic Polynomial ◽  
Complex Parameter ◽  
Rational Maps ◽  
Quadratic Maps ◽  
The Family ◽  
The One ◽  
Holomorphic Correspondences ◽  
Connectedness Locus

Abstract In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences $$\mathcal {F}_a$$Fa: $$\begin{aligned} \left( \frac{aw-1}{w-1}\right) ^2+\left( \frac{aw-1}{w-1}\right) \left( \frac{az+1}{z+1}\right) +\left( \frac{az+1}{z+1}\right) ^2=3 \end{aligned}$$aw-1w-12+aw-1w-1az+1z+1+az+1z+12=3and proved that for every value of $$a \in [4,7] \subset \mathbb {R}$$a∈[4,7]⊂R the correspondence $$\mathcal {F}_a$$Fa is a mating between a quadratic polynomial $$Q_c(z)=z^2+c,\,\,c \in \mathbb {R}$$Qc(z)=z2+c,c∈R, and the modular group $$\varGamma =PSL(2,\mathbb {Z})$$Γ=PSL(2,Z). They conjectured that this is the case for every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus. We show here that matings between the modular group and rational maps in the parabolic quadratic family $$Per_1(1)$$Per1(1) provide a better model: we prove that every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus is such a mating.

THE CONNECTEDNESS LOCUS OF A FAMILY OF REAL BIQUADRATIC POLYNOMIALS

2007 ◽  
Vol 17 (11) ◽  
pp. 4219-4222 ◽  
Author(s):  
YESHUN SUN ◽  
YONGCHENG YIN
Keyword(s):  
Real Numbers ◽  
Precise Description ◽  
The Family ◽  
Connectedness Locus

In this paper we present a precise description of the connectedness locus of the family of polynomials (z2 + x)2 + y, where x, y are real numbers.

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.

A STUDY OF THE DYNAMICS OF λ sin z

2002 ◽  
Vol 12 (12) ◽  
pp. 2869-2883 ◽  
Author(s):  
PATRICIA DOMÍNGUEZ ◽  
GUILLERMO SIENRA
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Julia Set ◽  
Parabolic Type ◽  
Fatou Set ◽  
Hyperbolic Component ◽  
Q Cycle ◽  
The Family ◽  
Fatou Components ◽  
Connectedness Locus

This paper studies the dynamics of the family λ sin z for some values of λ. First we give a description of the Fatou set for values of λ inside the unit disc. Then for values of λ on the unit circle of parabolic type (λ = exp (i2πθ), θ = p/q, (p, q) = 1), we prove that if q is even, there is one q-cycle of Fatou components, if q is odd, there are two q cycles of Fatou components. Moreover the Fatou components of such cycles are bounded. For λ as above there exists a component Dq tangent to the unit disc at λ of a hyperbolic component. There are examples for λ such that the Julia set is the whole complex plane. Finally, we discuss the connectedness locus and the existence of buried components for the Julia set.

scholarly journals CHEBYSHEV POLYNOMIALS ON JULIA SETS AND EQUIPOTENTIAL CURVES FOR THE FAMILY P(z)=zd−c

2009 ◽  
Vol 86 (2) ◽  
pp. 279-287
Author(s):  
YINGQING XIAO ◽  
WEIYUAN QIU
Keyword(s):  
Chebyshev Polynomials ◽  
Julia Set ◽  
Julia Sets ◽  
The Family

AbstractIt is shown that the dnth Chebyshev polynomials on the Julia set JP, and on the equipotential curve ΓP(R), of the polynomial P(z)=zd−c, are identical and exactly equal to the nth iteration of P(z) itself. As an application, the capacity of the Julia set JP is deduced, a result that was first obtained by Brolin.

DYNAMICS OF A FAMILY OF QUADRATIC MAPS IN THE QUATERNION SPACE

2005 ◽  
Vol 15 (08) ◽  
pp. 2535-2543 ◽  
Author(s):  
SHIZUO NAKANE
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set ◽  
Quadratic Maps ◽  
Quaternion Space ◽  
Connectedness Locus

The dynamics of a family of quadratic maps in the quaternion space is investigated. In particular, connectivity of the filled-in Julia sets is completely determined. It is shown that the connectedness locus of this family is not equal to what we call the quaternionic Mandelbrot set. Hyperbolic components will also be completely characterized.

Connectedness of the tricorn

1993 ◽  
Vol 13 (2) ◽  
pp. 349-356 ◽  
Author(s):  
Shizuo Nakane
Keyword(s):  
The Family ◽  
Connectedness Locus

AbstractIn this note, we show the connectedness of the tricorn, the connectedness locus for the family of antiquadratic maps: fc(z) = + c, c ∈ C.

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