Before studying very complicated cases, the simplest way to really understand what happends in an IFS, is to take a look at the simplest system that can be found: an IFS with only one function.
Actually, as there is only one function, this particular case is a deterministic problem which is in fact very classical.
The equation of evolution of the system can then be simply written:
(X(n+1)-B)=r*exp(ja)*(X(n)-B)
and we can then conclude that:
(X(n)-B)=r^{n}*exp(jna)*(X(0)-B)
The nth iterate is obtained by applying n times an homothetia
of ratio r and a rotation of angle a, both centered on B.
We can clearly see that the system is totally deterministic,
as the nth iterate only depends on the discrete time n, the parameter c=(r,a)
and the initial condition.
It is now clearly obvious that three major behaviour can
be distinguished:
(X(n+q)-B)=exp(jqa)*(X(n)-B)=exp(j*2pi*p)*(X(n)-B)=(X(n)-B).
Thus, every iterate X(n) is located on a set called an orbit of order q and which is defined by:
O = { X(k) such as (X(k)-B)=exp(j*k*2pi*(p/q))*(X(0)-B),
with k from 0 to (q-1) }.
+ If a is not the product of pi by a rationnal number, the successive iterates X(n) describe the whole circle of center B and of radius 1.
With this definition we can say that:
* In the first case ( | r | > 1 ), the iterates become larger and larger in modulus when n increases and so, they clearly does not respect the first condition: the system does not have an attractor.
* In the second case ( | r | < 1 ), the set A={B} is clearly the attractor of the system.
* When | r | = 1, we must pay a little
much more attention to determine if the system has an attractor:
+ In the first case, one may think that A=O. However,
an orbit O explicitly depends on the initial condition X(0) and so forth
cannot be considered as the attractor of the system.
+ In the second case, for any value of X(0), the iterates
X(n) will travel on the whole circle of radius 1 centered on B. This circle
respects the two conditions recquired above and thus is the attractor of
the system.
After this brief overview of the most simple IFS we can already make a few comments: