A contraction mapping on a metric space 
is a map 
for which there is a constant 
satisfying
 |
for all 
. Thus every image distance
is at most 
times the original distance. The Banach
fixed point theorem gives a unique fixed point
when 
is a nonempty complete
metric space. For example, 
on the real line has contraction
factor 
and fixed point 0. A nonempty finite collection of contraction mappings on a common compact metric space
forms an iterated function system (Hutchinson
1981).
