given by the matrix (both states are stationary). A periodic chain The Markov chain on given by the matrix is irreducible but periodic. Thus it is not ergodic in the sense of Markov chain though the associated measure on is ergodic for the shift map. However the shift is not mixing for this measure, as for the sets and we have but Generalisations The definition of ergodicity also makes sense for group actions. The classical theory (for invertible transformations) corresponds to actions of or . For non-abelian groups there might not be invariant measures even on compact metric spaces. However the definition of ergodicity carries over unchanged if one replaces invariant measures by quasi-invariant measures. Important examples are the action of a semisimple Lie group (or a lattice therein) on its Furstenberg boundary. A measurable equivalence relation it is said to be ergodic if all saturated subsets are either null or conull. Notes References External links Karma Dajani and Sjoerd Dirksin, "A Simple Introduction to Ergodic Theory" Ergodic theory