Blowup points and baby Mandelbrot sets for singularly perturbed rational maps

2006 ◽  
pp. 51-62 ◽  
Author(s):  
Robert L. Devaney ◽  
Matt Holzer ◽  
David Uminsky
Keyword(s):  
Singularly Perturbed ◽  
Rational Maps ◽  
Mandelbrot Sets

Generalized baby Mandelbrot sets adorned with halos in families of rational maps

2016 ◽  
Vol 23 (3) ◽  
pp. 503-520 ◽  
Author(s):  
HyeGyong Jang ◽  
YongNam So ◽  
Sebastian M. Marotta
Keyword(s):  
Rational Maps ◽  
Mandelbrot Sets

The escape trichotomy for singularly perturbed rational maps

2005 ◽  
Vol 54 (6) ◽  
pp. 1621-1634 ◽  
Author(s):  
Robert L. Devaney ◽  
Daniel M. Look ◽  
David Uminsky
Keyword(s):  
Singularly Perturbed ◽  
Rational Maps

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 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.

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.

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