Endomorphism

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 .

REFERENCES:

Lang, S. Algebra, rev. 3rd ed. New York: Springer-Verlag, 2002.