If φ is a map then a measure μ is invariant under φ if μ(φ − 1A) = μ(A) for all A. In the context of Hamiltonian flows, an invariant measure on phase space is invariant under the Hamiltonian flow map. This measure is a probability measure if it is positive and has total measure of size 1.
Examples of invariant measures
- Gibbs measure
- Every invariant torus supports at least one invariant measure.