Invariant measures

If $\phi$ is a map then a measure $\mu$ is invariant under $\phi$ if $\mu (\phi ^{-1}A)=\mu (A)$ 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.