Proof of the Ergodic Theorem.¶
Birkhoff, G. D. (1931). Proof of the Ergodic Theorem. Proceedings of the National Academy of Sciences USA, 17(12), 656-660.
Cited by¶
3 citations across 3 artifacts.
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Primes¶
- Chaos
- Their unpredictability is individual-trajectory, not ensemble; the long-time average over a single trajectory equals the space-average over the attractor (Birkhoff's (1931) ergodic theorem
This sourcePointwise ergodic theorem establishing that for measure-preserving transformations time averages along almost every trajectory equal the space average; foundational for treating chaotic dynamics via invariant measures.
- Their unpredictability is individual-trajectory, not ensemble; the long-time average over a single trajectory equals the space-average over the attractor (Birkhoff's (1931) ergodic theorem
- Ensemble
- . Birkhoff's ergodic theorem
This sourceThe pointwise ergodic theorem — for measure-preserving transformations, time averages along almost every trajectory equal the space (ensemble) average — justifying ensemble replacement of single trajectories for ergodic systems.
- . Birkhoff's ergodic theorem
Domain-specific¶
Verification¶
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