Properties
Hutchinson (1981) showed that, for the metric space, such a system of functions has a unique compact (closed and bounded) fixed set S. One way of constructing a fixed set is to start with an initial point or set S0 and iterate the actions of the fi, taking Sn+1 to be the union of the image of Sn under the fi ; then taking S to be the closure of the union of the Sn. Symbolically, the unique fixed (nonempty compact) set has the property
The set S is thus the fixed set of the Hutchinson operator
The existence and uniqueness of S is a consequence of the contraction mapping principle as is the fact that
for any nonempty compact set in . Random elements of S may be obtained by the "chaos game" below.
The collection of functions generates a monoid under composition. If there are only two such functions, the monoid can be visualized as a binary tree, where, at each node of the tree, one may compose with the one or the other function (i.e. take the left or the right branch). In general, if there are k functions, then one may visualize the monoid as a full k-ary tree, also known as a Cayley tree.
Read more about this topic: Iterated Function System
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)