In order to investigate fractal graphs from quotient space theory, the concept of quotient fractals is established first. Then, the quotient fractals are used to approximate fractal graphs. Its procedure is the following. An equivalence relation is defined through a mapping. A corresponding chain of hierarchical quotient spaces is built by the equivalence
relation. Then, quotient mappings are set up on the quotient spaces. Finally, quotient fractals are obtained from the quotient mappings.
Theorem 7.4 (Attractor Theorem)
Assume that
is an iterated function system on
X.
W is a fractal mapping on
X. Then,
holds on (
H(
x),h(d)). Namely, there exists a unique fixed point
P on
H(
x), i.e.,
P is a corresponding fractal graph on iterated function system IFS=W.