R < 1
and both | r1 | and | r2 | are smaller than 1.

This case is the most interesting as we are sure that the system has an attractor which is a generalised 2-ratios Cantor set.
This generalised Cantor set is obtained as a classical Cantor set, but at each iteration, instead of replacing an interval of size D by two intervals of size r*D, this interval is replaced by intervals of size r1*D and r2*D.
It is also equivalent to say that the initial interval [B1;B2] is replaced by the two transformed sets h1([B1;B2]) and h2([B1;B2]) where h1 and h2 are the two pre-defined homothetias, and so on at each iteration.

The fractal dimension d of this set is such as:

| r1 |d + | r2 |d = 1.

To be more precise, it is likely to discuss 2 different cases:

In this case, it can be shown that the fractal dimension d defined above is greater than 1.
This is obvious when looking at the way the associated Cantor set is built at each iteration:
Cantor set for |r1|+|r2|>1

However, the system is single-dimensioned (as it must remain on the real axe) and thus, its euclidian dimension must be equal to 1.
In fact, this is not as paradoxal as it seems to be: the final attractor is an interval of the real axe and on this interval, the probability density function (pdf) for an iterate to be in the neighbourhood of a point is a fractal curve.

The exemple below shows this phenomenon more clearly with r1=0.7 and r2=0.35.

X(n) vs n

This graph shows the positions of the successive iterates at a given iteration.
As you can see, the iterates go through the whole interval [B1;B2].
The attractor is then [B1;B2] (this is only true when r1 and r2 are positive: else, the attractor is a segment [B1-b1;B2-b2] with b1 and b2 depending on the values of r1 and r2).

However, the structure of the fractal attractor is appearing in the background.
The problem is that this fractal attractor is too "dense" and that it fills the whole interval (its dimension is approximately d~1.085).

The next graph is the probability density of the iterates on this interval:

Density of probability of the iterates

Although the whole interval [-1;1] is the attractor of the system, the graph above shows that some part of this interval have a greater density than other parts. What is important is that the pdf never becomes nul: this is the result of the recovering of the fractal set with itself because it is confined into a set whose dimension is smaller than it's own fractal dimension.
This curve is fractal by itself (with a multi-fractal structure) and this is the real important point, as I will explain it further.

In this case, the attractor is effectively a fractal set whose dimension d is smaller than one.

The result of a simulation with r1=0.5 and r2=0.3 is plotted below:

Cantor set for |r1|+|r2|<1

We can see that the result is equivalent to the associated Cantor set of dimension d~ 0.75:
The interval D=[-1;1] has been replaced at the first iteration by [-1;0]U[0.4;1]=D1 U D2 and we have m(D1)=r1*m(D) and m(D2)=r2*m(D) with m([A;B])=B-A (m(I) is the measure of the interval I). When applying this scheme an inifinite number of time, the result is a Cantor set.

Here again, the shape of the attractor does not depends of p1 and p2.
However the pdf on the interval [-1;1] depends of p1 and p2 but also of r1 and r2.

In fact, the support of this function is the Cantor set above (which only depends of r1 and r2) but the value of the pdf (probability density function) depends of all these parameters.
It is very important to notice than once more, the pdf is a fractal curve.
Actually, this curve is only defined in terms of distributions: its support has a nul measure and its integral over the interval is equal to one.
Although it is not easy to deal with this situation in the case of the IFS formulation, the theories of weighted Cantor sets and of multifractals provides us a formalism that is perfectly adapted to the situation.
It will be too long to explain this here and interested readers should find more explanations in ref. [1].

What should be remembered is that in both cases, the pdf of the iterates is a multifractal curve that can be caracterised by its Hölder exponent (describing its' singularity) and its dimension distribution.


see you ...