# Invariant measures

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