# What is the condensation set of a fractal?

Is there a definition of the condensation set of a fractal that is both clear and rigorous?

I've been searching around to get a sense of what exactly the condensation set of a fractal is - I've encountered papers like this: http://www-users.cs.umn.edu/~bstanton/pdf/p271-demko.pdf

or an occasional mention like this: http://faculty.cs.tamu.edu/schaefer/teaching/441_Spring2012/assignments/fractals.html

I would say that a condensation set is not really part of the mathematical theory of fractal geometry proper; rather, it's technique used to enhance the images generated by the Iterated Function System (IFS) process. The invariant set of an IFS, for example, is uniquely defined by the IFS itself. The choice of a condensation set is somewhat arbitrary.

As far as a rigorous definition goes, I believe the concept was introduced by Michael Barnsley, a pioneer in fractal image compression. You can read his exposition in section 3.9 of his book Fractals Everywhere. His idea is to introduce a constant set valued function $f_0$ to a normal IFS. Note that $f_0$ is not an extension of a normal function on $\mathbb R^n$; rather it maps the set of compact sets to itself and assumes just one possible value. As a result, that set appears in the final attractor in a repeating fashion.

My opinion is that choice of this set is quite arbitrary and adds nothing of mathematical value to the attractor of the original IFS, though it can certainly lead to some nice visual effects. This same effect can be achieved by choosing your initial seed for the IFS to be the condensation set and retaining that set, and its images as you iterate the IFS.

Here's an example lifted from this question. Consider the IFS consisting of the following three functions: \begin{align} f_1(x,y) &= \frac{1}{2}R\left(\frac{\pi}{2}\right) \left( \begin{array}{c} x \\ y \end{array} \right) + \left( \begin{array}{c} 1/2 \\ 0 \end{array} \right) \\ f_2(x,y) &= \frac{1}{2} \left( \begin{array}{c} x \\ y \end{array} \right)+ \left( \begin{array}{c} 0 \\ 1/2 \end{array} \right) \\ f_2(x,y) &= \frac{1}{2}R\left(-\frac{\pi}{2}\right) \left( \begin{array}{c} x \\ y \end{array} \right)+ \left( \begin{array}{c} 1/2 \\ 1 \end{array} \right), \\ \end{align} where $R(\theta)$ is the $2\times2$ rotation matrix through the angle $\theta$. The image of a set $A$ under this IFS is simply $$\bigcup_{i=1}^3 f_i(A).$$ An oriented unit square, together with it's image under this IFS, looks like so:

If you start with a solid unit square and apply the IFS iteratively, you generate a sequence of images that looks like so:

If we choose a small, rotated square centered at $(1/2,1/2)$ as a codensation set, we generate a sequence of images like so:

And, after a few more iterates: