Help with fractals

Question

Let $f(z)=z^2+4z+1$. Is the filled Julia set (denoted $F_f$) connected?

I'm not sure to show how its connected. The only thing I know how to do is verify whether a given point is in the set.

Answer 1

You only need to test the critical point, the root of the derivative $2z+4$, that is, $z=-2$. You'll find that the orbit of $z=-2$ is $-2,-3,-2,-3,\dots$, periodic of period 2. Since the critical orbit does not goes to infinity, the Julia set is connected.

Alternatively, you reduce $z^2+4z+1$ to the form $z^2+c$ by conjugation with a map $z\mapsto z+a$ for some $a$. For this, you need to find $a$ and $c$ such that $(z+a)^2+c-a=z^2+4z+1$. You'll find that $a=2$ and $c=-1$. So, the Julia set of $z^2+4z+1$ corresponds to the Julia set of $z^2-1$, which is connected.

(image from http://www.math.cornell.edu/~lipa/mec/lesson5.html)