for some real constant 0<= *c* <1, the *contractivity factor*.

**Theorem:** (The Contraction Mapping Theorem) Let
*f*:*X*->*X* be a contraction mapping on a complete metric space
(*X*,*d*). Then *f* has exactly one fixed point,
*a*, in *X*, and:

Contraction mappings are the elementary building blocks of IFSs, but they are un-interesting by themselves (as seen by the above theorem).

where each *w _{n}* is a contraction mapping. We define
the contractivity factor of the system to be:

Here *c _{n}* is the contraction factor for

**Def:** Given a metric space (*X*,*d*), we define
another metric space (*H*(*X*),*h*(*d*)). Where
*H*(*X*) is the set of all nonempty compact subsets of
*X*, and *h*(*d*) is the Hausdorf distance between two
elements of *H*(*X*). In rough terms, the Hausdorff
distance between two sets *A* and *B* is *d* if every
point of *A* is within *d* of some point of *B* and
vice versa.

This metric space (*H*,*h*) is in some sense the natural
space in which fractals live. In a general way a fractal is an element
of this space. However, this is a mathematical abstraction of our
intuitive ideas about what a fractal should be (since *H* includes
lot's of normal geometric objects as well.)

**Theorem:** Let {*X*; *w _{n}*,

where

is a contraction mapping on (*H*(*X*),*h*(*d*))
with contractivity factor *s*. Further, the unique fixed point,
*A*, of *W* is given by:

Note that a point of *H* is actually a non-empty compact set of
the original space *X*. This set *A* is called the
*attractor* of the IFS.

It is the attractors of IFSs, which live in *H*(*X*), which
are really fractals. Indeed, almost all of the well known fractals,
as well as many less well known ones, are the attractors of
appropriate IFSs. See some examples
generated using Fractalina.

Below is an applet that implements the deterministic algorithm for the IFS:

Notice that the initial set can vary widely, but the result converges rapidly to the attractor of the IFS. (The attractor of this IFS is the well known Sierpinski triangle.)

To chose the initial set drag in the window, then use the Iterate button to
step through iterations on that set.

This theorem provides the mathematical basis for animations of IFSs (in particular for Franimate!). The effect implied here has been called "blowing in the wind", because an imaginary fractal tree can be made to blow in an imaginary mathematical breeze, by continuously varying a parameter to the IFS of the "tree".

*Fractals Everywhere*, Michael Barnsley, Academic Press Inc., 1988.

This page was created by Noah
Goodman.

Comments to:
webmaster@geom.umn.edu

Created: Sep 23 1996 ---
Last modified: Tue Oct 8 11:30:11 1996