Ergodic Measure

An Endomorphism is called ergodic if it is true that $T^{-1}A = A$ Implies $m(A) = 0$ or 1, where $T^{-1}A = \{ x \in X : T(x) \in A \}$. Examples of ergodic endomorphisms include the Map $X \rightarrow 2x$ mod 1 on the unit interval with Lebesgue Measure, certain Automorphisms of the Torus, and ``Bernoulli shifts'' (and more generally ``Markov shifts'').

Given a Map $T$ and a Sigma Algebra, there may be many ergodic measures. If there is only one ergodic measure, then $T$ is called uniquely ergodic. An example of a uniquely ergodic transformation is the Map $x \mapsto x + a$ mod 1 on the unit interval when $a$ is irrational. Here, the unique ergodic measure is Lebesgue Measure.