Generating Fractal Equations

Date: 7/1/96 at 22:26:3
From: Anonymous
Subject: Generating Fractal Equations

Is there any way to generate a fractal equation from a set of numbers?
For example, taking a million or so seemingly random numbers and 
"reversing" the fractal process to provide a formula that will 
generate the same series of numbers every time?

Date: 7/7/96 at 0:43:6
From: Doctor Pete
Subject: Re: Generating Fractal Equations

In short, the answer is no.

Before I explain in further detail, let me set down some preliminary
notions; these you may or may not already be familiar with.

I am assuming what you refer to as "fractal process" is the iteration 
of some mapping such as z |--> f(z).  That is, we start with an 
initial value z0, and calculate the values f(z0), f(f(z0)), 
f(f(f(z0))), ... and plot these iterates.  The set of iterates under 
the mapping defined above is called the *forward orbit* (or for 
simplicity, orbit) of z0 under the mapping z |--> f(z).  The *inverse 
orbit* or *backward orbit* is the set of iterates of the inverse 
mapping, z |--> f^-1(z), where f^-1(f(z))=z.  Note that such a mapping 
may be multivalued, such as in the case where f(z)=z^2+c; another term 
for the iterates of the inverse mapping of f is the *preimages* of z0 
under f.

So this is basically how we generate sets of points (and pretty 
pictures). Take some function f, and a point z0, and we plot the orbit 
of z0 under f. There are other ways of plotting points, such as 
numerical solution of differential equations, but in general the 
concept is the same:  we use iteration to get the next point from the 
previous.

Now for the explanation:

One of the things about iteration is that it has the property of 
magnifying errors - that is, if two points z1 and z2 are initially 
very close together, their difference will increase with every 
iteration.  This is also the exact same property that gives fractals 
their complex and unpredictable nature. So after, say, fifteen 
iterations, the roundoff error created by the computer will be too 
large to make the value of the point have any specific meaning; that 
is, if you calculated 50 iterations, and then take 50 inverse 
iterations, you will not get your original point back.  This is why 
you cannot take a set of data and represent it by a mapping if the 
data is fractal. 

However, if you knew the order in which the points were plotted, you 
have orbit information which *might* let you *approximate* the data 
with some sort of mapping.  But this relies on understanding the 
dynamics of the mapping beforehand, which is highly unlikely.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/