These case is a little bit hard to discuss because it shows a "pathological" behaviour.
As we have seen it previously, we have R<1 which means
that the probability for the system to diverge is equal to zero.
However, what is important for us is the behaviour of
the system after an infinite number of iteration.
Hence, after a given number of iteration, the probability
for |X(n)| to be greater than a given number A is decreasing to zero when
A increases.
But if we wait long enough, although the probability
for |X(n)| to reach any given large number tends to zero, this event can
yet happends...
This paradoxal situation can be made more understandable
on the next graph.
For this simulation, we have r_{1}=1.4 and r_{2}=0.60
(hence R<1):
We can see that most of the time, the iterates are very
close to the origin, but that suddenly, when the repulsive center is choosen
successively a certain number of times, the iterates becomes larger and
larger.
The problem is that even if the probability of the repulsive
center (p_{1} for us) is very small, if we wait long enough, this
repulsive center will be choosen successively as many times as we want.
The system can then diverge to infinity, but only with
a probability equal to zero.....
The graph below shows the probability density of the iterates
on the interval [-2;18]:
First of all, one must notice that this graph seems to
be discontinuous: this is only the consequence of the finite number of
iterations and of the computation method to obtain this density.
Though, it is clearly visible that this density is maximal
for a finite value (that depends of r_{1}, r_{2}, p_{1}
and p_{2}) and that it tends to zero when X(n) increases.
One may think that it might then be possible to say that the whole real axe (or here the set of the positive real numbers) is the attractor of the system.
According to me, it is not a good definition and this class of system has no attractor.
Actually, to understand why, I should precise that if a system is described by n independant variables, we can say that it is a n-dimension system.
Moreover, we should define another concept which is the
distance between a set A and a point X:
The distance d(X,A) of X to A is equal to the minimum
of the n-dimension euclidian distance d_{n}(X,Y) between the point
X and any point Y in A.
With theses definitions, I would say that a set A can be considered as the attractor of a n-dimension system if:
Conclusion:
This type of behaviour is very particular and can be
seen as a random ditribution with special momentum laws.
It is also important to notice that when the number of
iterations increases, this pdf has a well-defined limit (this will no longer
be true in the next part).
However, with this precise definition, we can say that
this class of IFS has no attractor and we can now turn ourselves to the
most interesting cases (where there is an attractor) ....