Ergodic Measure

An endomorphism is called ergodic if it is true that  implies  or 1, where . Examples of ergodic endomorphisms include the map  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  and a sigma-algebra, there may be many ergodic measures. If there is only one ergodic measure, then  is called uniquely ergodic. An example of a uniquely ergodic transformation is the map  mod 1 on the unit interval when  is irrational. Here, the unique ergodic measure is Lebesgue measure.