The term endomorphism derives from the Greek adverb endon ("inside") and morphosis ("to form" or "to shape").
In algebra, an endomorphism of a group, module, ring, vector space, etc. is a homomorphism from one object to itself (with surjectivity not required).
In ergodic theory, let 
be a set, 
a sigma-algebra on 
and 
a probability measure.
A map 
is called an endomorphism (or measure-preserving transformation)
if
1. 
is surjective,
2. 
is measurable,
3. 
for all 
, where 
denotes the pre-image
of 
.
An endomorphism is called ergodic if it is true that 
implies 
or 1, where 
.
