Endomorphism

A Surjective Morphism from an object to itself. In Ergodic Theory, let $X$ be a Set, $F$ a Sigma Algebra on $X$ and $m$ a Probability Measure. A Map $T: X \rightarrow X$ is called an endomorphism or Measure-Preserving Transformation if

1. $T$ is Surjective,

2. $T$ is Measurable,

3. $m(T^{-1}A) = m(A)$ for all $A \in F$.

An endomorphism is called Ergodic if it is true that $T^{-1}A = A$ Implies $m(A) = 0$ or 1, where $T^{-1}A = \{ x \in X : T(x) \in A \}$.

See also Measurable Function, Measure-Preserving Transformation, Morphism, Sigma Algebra, Surjective