Fatou components and Julia sets of singularly perturbed rational maps with positive parameter

2012 ◽  
Vol 28 (10) ◽  
pp. 1937-1954 ◽  
Author(s):  
Wei Yuan Qiu ◽  
Lan Xie ◽  
Yong Cheng Yin
Keyword(s):  
Julia Sets ◽  
Singularly Perturbed ◽  
Rational Maps ◽  
Positive Parameter ◽  
Fatou Components

Limiting behavior of Julia sets of singularly perturbed rational maps

2014 ◽  
Author(s):  
Robert L. Devaney
Keyword(s):  
Singular Perturbations ◽  
Julia Sets ◽  
Singularly Perturbed ◽  
Rational Maps ◽  
Rational Map ◽  
Limiting Behavior ◽  
Jordan Curves ◽  
Hyperbolic Component ◽  
Dynamical Properties ◽  
Sierpinski Gaskets

This chapter surveys dynamical properties of the families fsubscript c,𝜆(z) = zⁿ + c + λ‎/zᵈ for n ≥ 2, d ≥ 1, with c corresponding to the center of a hyperbolic component of the Multibrot set. These rational maps produce a variety of interesting Julia sets, including Sierpinski carpets and Sierpinski gaskets, as well as laminations by Jordan curves. The chapter describes a curious “implosion” of the Julia sets as a polynomial psubscript c = zⁿ + c is perturbed to a rational map fsubscript c,𝜆. In this way the chapter shows yet another way of producing rational maps through “singular” perturbations of complex polynomials.

On the dynamics of generalized McMullen maps

2013 ◽  
Vol 34 (6) ◽  
pp. 2093-2112 ◽  
Author(s):  
YINGQING XIAO ◽  
WEIYUAN QIU ◽  
YONGCHENG YIN
Keyword(s):  
Critical Points ◽  
Julia Sets ◽  
Dynamical Behavior ◽  
Parameter Family ◽  
Rational Maps ◽  
Topological Description ◽  
Image Position ◽  
Fatou Components ◽  
Two Parameter

AbstractIn this paper, we study the dynamics of the two-parameter family of rational maps $$\begin{eqnarray*}{F}_{a, b} (z)= {z}^{n} + \frac{a}{{z}^{n} } + b.\end{eqnarray*}$$ We give the topological description of Julia sets and Fatou components of ${F}_{a, b} $ according to the dynamical behavior of the orbits of its free critical points.

Density of hyperbolicity for rational maps with Cantor Julia sets

2011 ◽  
Vol 32 (5) ◽  
pp. 1711-1726 ◽  
Author(s):  
WENJUAN PENG ◽  
YONGCHENG YIN ◽  
YU ZHAI
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

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