Dynamical systems and ergodic theory Ergodic theory Concept
Invariant measure
Summary
An invariant measure of a map is a measure that the map does not change: every set has the same measure as its preimage. With respect to such a measure the map is a measure-preserving transformation, and the tools of ergodic theory apply. Invariant probability measures describe the statistics of orbits, such as how often an orbit visits a region. On a compact metric space every continuous map has at least one invariant probability measure, by the Krylov–Bogolyubov theorem.
Where this leads
19 pages build on this one.
- Krylov–Bogolyubov theorem
- Ergodic decomposition
- Feigenbaum–Cvitanović functional equation
- Milnor–Thurston kneading theory
- Crisis
- Parry measure
- Tent map
- Feigenbaum constants
- Intermittency
- Dynamical billiards
- Variational principle for topological entropy
- Bifurcation diagram
- Unique ergodicity
- Gibbs measure
- Logistic map
- Pesin theory
- Jakobson's theorem
- Sinai–Ruelle–Bowen measure
- Singer's theorem
Further reading
About this page
| Tag | B232EZ1FMB |
|---|---|
| URI | m2://B232EZ1FMB |
| Kind | concept |
| Writing | W0 stub |
| MSC | 37A05, 28D05, 37C40 |
| Same as |
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Used by
- Krylov–Bogolyubov theorem
- Ergodic decomposition
- Parry measure
- Tent map
- Dynamical billiards
- Variational principle for topological entropy
- Unique ergodicity
- Gibbs measure
- Pesin theory
- Jakobson's theorem
- Sinai–Ruelle–Bowen measure