Birkhoff ergodic theorem

The Birkhoff ergodic theorem, proved by George Birkhoff in 1931, is the foundational result of ergodic theory. It states that for a measure-preserving dynamical system, the time average of an integrable function along almost every orbit equals the space average of the function with respect to the invariant measure. This theorem transforms the question of whether a system is statistically predictable into a question about the existence and properties of invariant measures.

The Birkhoff theorem is the additive counterpart to the multiplicative Oseledets theorem. Where Birkhoff concerns the asymptotic behavior of scalar observables, Oseledets concerns the asymptotic behavior of matrix-valued cocycles. The progression from Birkhoff to Oseledets is the progression from classical ergodic theory to the spectral theory of chaotic systems.