Dynamics of the quadratic rational maps

Abstract

We study the dynamics of quadratic rational maps and the connectedness of its Julia sets. Any quadraitc rational map is conjugate to either $z^2+c$ or $\lambda(z+1/z)+b$. For $\mid \lambda \mid = 1$, we characterize the Mandelbrot set $M \lambda$, the set of parameters b for which the Julia set of $\lambda(z+1/z)+b$ is connected. It is seen to be the whole complex plane if $\lambda \neq 1$, but it is an intricate fractal if $\lambda = 1$. This extends the previous work done for the case $\mid \lambda \mid>1$. We also give some properties of the dynamics of the map $z+1/z+b$ and its Mandelbrot set $M_1$, and present algorithms for drawing the Mandelbrot set and the Julia sets by using computer graphics techniques.