Consider the following picture:
I made this this image the following way:
While computing $z_{n+1} = z_n^2 + c$ with $z_0 = 0$ and $c$ being the point on the complex plane, check if $|z_{n+1}| < |z_n|$. If so, increase a counter $D$. Repeat until $n > N_{max}$ or $|z_n| > 4$, where $N_{max}$ is the iteration limit.
If the iteration stops because $n > N_{max}$, then set $l = \frac{D}{N_{max}}$. If the iteration stops because $|z_n| > 4$, then set $l = D$
By plotting the value of $l$ for every point in the complex plane, one obtains the image above.
(Basically I am plotting how many times the absolute value of $z_{n+1}$ decreases compared to $z_n$) (This works for julia-sets in a similar fashion.)
The interior of the main cardiod gets pretty messy, but everything else seems fine.
Now I have a question: Is this a known method?
I have got a second question regarding the image: At the value $-1.543689012692076361570855971801569$ the image shows a wierd cusp, and the corresponding Julia-set is a dendrite. Is this somehow related?
