The connectedness locus of the family of polynomialsPc(z) = zn–cz

2013 ◽  
Vol 58 (1) ◽  
pp. 99-107
Author(s):  
Carlos Arteaga ◽  
Alexandre Alves
Keyword(s):  
The Family ◽  
Connectedness Locus

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.

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.

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.

Triassic sponges (Sphinctozoa) from Hells Canyon, Oregon

1988 ◽  
Vol 62 (03) ◽  
pp. 419-423 ◽  
Author(s):  
Baba Senowbari-Daryan ◽  
George D. Stanley
Keyword(s):  
Endemic Species ◽  
Upper Triassic ◽  
Island Arc ◽  
Tropical Island ◽  
The Family ◽  
Wallowa Terrane ◽  
Unique Condition ◽  
Hells Canyon

Two Upper Triassic sphinctozoan sponges of the family Sebargasiidae were recovered from silicified residues collected in Hells Canyon, Oregon. These sponges areAmblysiphonellacf.A. steinmanni(Haas), known from the Tethys region, andColospongia whalenin. sp., an endemic species. The latter sponge was placed in the superfamily Porata by Seilacher (1962). The presence of well-preserved cribrate plates in this sponge, in addition to pores of the chamber walls, is a unique condition never before reported in any porate sphinctozoans. Aporate counterparts known primarily from the Triassic Alps have similar cribrate plates but lack the pores in the chamber walls. The sponges from Hells Canyon are associated with abundant bivalves and corals of marked Tethyan affinities and come from a displaced terrane known as the Wallowa Terrane. It was a tropical island arc, suspected to have paleogeographic relationships with Wrangellia; however, these sponges have not yet been found in any other Cordilleran terrane.

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