Dynamical systems and ergodic theory Ergodic theory Concept
Ergodic transformation
Summary
A measure-preserving transformation of a probability space is ergodic if it has no invariant measurable set other than null sets and sets of full measure. Ergodic maps cannot be split into two separate systems of positive measure. By Birkhoff's ergodic theorem, for an ergodic map the time average of an integrable function along almost every orbit equals its space average. This is the mathematical form of the ergodic hypothesis of statistical mechanics.
Where this leads
13 pages build on this one.
Further reading
About this page
| Tag | AZRPHS478B |
|---|---|
| URI | m2://AZRPHS478B |
| Kind | concept |
| Writing | W0 stub |
| MSC | 37A25, 28D05 |
| Specializes | Measure-preserving transformation |
Markdown · JSON · Edit on GitHub
What this page needs for W1
- an inventory row: the row of the inventory that the node accounts for (0 of 1)
- a source with a locator (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Results about this
Used by
- Ergodic decomposition
- Ergodic hypothesis
- Kingman's subadditive ergodic theorem
- Quasi-ergodicity
- Sinai's factor theorem
- Halmos–von Neumann theorem
- Quantum ergodicity
- Shannon–McMillan–Breiman theorem
- Birkhoff's ergodic theorem
- Kac's lemma
- Oseledets theorem