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 r_{1}*D and r_{2}*D.
It is also equivalent to say that the initial interval
[B_{1};B_{2}] is replaced by the two transformed sets h_{1}([B_{1};B_{2}])
and h_{2}([B_{1};B_{2}]) where h_{1} and
h_{2} are the two pre-defined homothetias, and so on at each iteration.
The fractal dimension d of this set is such as:
To be more precise, it is likely to discuss 2 different
cases:
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 r_{1}=0.7 and r_{2}=0.35.
This graph shows the positions of the successive iterates
at a given iteration.
As you can see, the iterates go through the whole interval
[B_{1};B_{2}].
The attractor is then [B_{1};B_{2}] (this
is only true when r_{1} and r_{2 }are positive: else, the
attractor is a segment [B_{1}-b_{1};B_{2}-b_{2}]
with b_{1} and b_{2} depending on the values of r_{1}
and r_{2}).
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:
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.
The result of a simulation with r_{1}=0.5 and r_{2}=0.3 is plotted below:
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]=D_{1} U D_{2} and we have m(D_{1})=r_{1}*m(D)
and m(D_{2})=r_{2}*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 p_{1} and p_{2}.
However the pdf on the interval [-1;1] depends of p_{1}
and p_{2} but also of r_{1} and r_{2}.
In fact, the support of this function is the Cantor set
above (which only depends of r_{1} and r_{2}) 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.