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

Abstract

We consider the family of rational maps given by $F_{\mathit{\lambda }}(z)=z^n+\mathit{\lambda }/z^d$ where $n,d\in \mathbb{N}$ with $1/n+1/d<1,$ the variable $z\in \widehat{C}$ and the parameter $\mathit{\lambda }\in \mathbb{C}$. It is known [1] that when $n=d\ge 3$ there are $n-1$ small copies of the Mandelbrot set symmetrically located around the origin in the parameter $\mathit{\lambda }-$plane. These baby Mandelbrot sets have ‘antennas’ attached to the boundaries of Sierpiński holes. Sierpiński holes are open simply connected subsets of the parameter space for which the Julia sets of $F_{\mathit{\lambda }}$ are Sierpiński curves. In this paper we generalize the symmetry properties of $F_{\mathit{\lambda }}$ and the existence of the $n-1$ baby Mandelbrot sets to the case when $1/n+1/d<1$ where n is not necessarily equal to d.

Keywords:

• Julia sets
• Mandelbrot set
• rational maps
• complex dynamics

AMS Subject Classifications:

• 37f10

Notes

No potential conflict of interest was reported by the authors.

1 We use $\mathbb{C}$ for the complex plane and $\widehat{C}=\mathbb{C}\cup \{\infty \}$ for the Riemann sphere.

Additional information

Funding

Dr Jang has written this paper during a visit of the CAS supported by the TWAS-UNESCO Associateship Scheme.