IFS Fractal

First steps:
1 complex function.

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)=rⁿ*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:

If | r | > 1, then rⁿ diverges and of course X(n) also diverges.

If | r | < 1, then rⁿ converges to zero and X(n) converges to B.

If | r | = 1, two situations arise:

+ If we can find two numbers p and q that are prime with one another such as a=(p/q)*2pi, we can write that:

(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) }.

Orbit with p/q=5/24

+ 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.

Circle attractor

We can already see that a main concept is appearing in these very simple cases: the existence of an attractor for the system. In fact, it is not obvious to give a clear definition to this concept at this point of the discussion and for now, we might just say that the set A is the attractor of a system if:
* the modulus of X(n) remains finite when n becomes very large.
* for any initial condition X(0) and when n becomes very large, the "distance" between X(n) and A becomes very small (the distance between a point X and a set A will be defined further, but is quite intuitive).

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:

This very simple case gives us a first idea of the general behaviour of any IFS. Actually, at each iteration, the evolution of any multi-function IFS is governed by the equations above and any IFS can be seen as the result of a competition between each function.
We have been able to sort all the previous systems according to their attractor and one of the main issue in the study of general IFS will certainly be to guess if the system has an attractor and what kind of attractor (shape, size, dimension, ...) is associated to a particular set of parameter.