volume measure); the horocycle flow on a hyperbolic manifold of finite volume is ergodic (for the normalised volume measure) Ergodicity in compact metric spaces If is a compact metric space it is naturally endowed with the σ-algebra of Borel sets. The additional structure coming from the topology then allows a much more detailed theory for ergodic transformations and measures on . Functional analysis interpretation A very powerful alternate definition of ergodic measures can be given using the theory of Banach spaces. Radon measures on form a Banach space of which the set of probability measures on is a convex subset. Given a continuous transformation of the subset of -invariant measures is a closed convex subset, and a measure is ergodic for if and only if it is an extreme point of this convex. Existence of ergodic measures In the setting above it follows from the Banach-Alaoglu theorem that there always exists extremal points in . Hence a transformation of a compact metric space always admits ergodic measures. Ergodic decomposition In general an invariant measure need not be ergodic, but as a consequence of Choquet theory it can always be expressed as the barycenter of a probability measure on the set of ergodic measures. This is referred to as the ergodic decomposition of the measure. Example In the case of and the counting measure is not ergodic. The ergodic measures for are the uniform measures supported on the subsets and and every -invariant probability measure can be written in the form for some . In particular is the ergodic decomposition of the counting measure. Continuous systems Everything in this section transfers verbatim to continuous actions of or on compact metric spaces. Unique ergodicity The transformation is said to be uniquely ergodic if there is a unique Borel probability measure on which is ergodic for . In the examples considered above, irrational rotations of the circle are uniquely ergodic; shift maps are not. Probabilistic interpretation: ergodic processes If is a discrete-time stochastic process on a space , it is said to be ergodic if the joint distribution of the variables on is invariant under the shift map . This is a particular case of the notions discussed above. The simplest case is that of an independent and identically distributed process which corresponds to the shift map described above. Another important case is that of a Markov chain which is discussed in detail