Editing Julia set (section)

===Using backwards (inverse) iteration (IIM) ===
[[Image:Reversed Julia set C = ( 0.4 0.3 ).gif|thumb|A Julia set plot, generated using random IIM]] 
[[Image:Miimcr.png|right|thumb|A Julia set plot, generated using MIIM]] 
As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point ''z'' we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of ''z'' under some high iterate <math>f^n</math> of ''f''.

Unfortunately, as the number of iterated pre-images grows exponentially, this is not feasible computationally. However, we can adjust this method, in a similar way as the "random game" method for [[iterated function system]]s. That is, in each step, we choose at random one of the inverse images of ''f''.

For example, for the quadratic polynomial ''f<sub>c</sub>'', the backwards iteration is described by

:<math>z_{n-1} = \sqrt{z_n - c} .</math>

At each step, one of the two square roots is selected at random.

Note that certain parts of the Julia set are quite difficult to access with the reverse Julia algorithm. For this reason, one must modify IIM/J ( it is called MIIM/J) or use other methods to produce better images.