Invariant measures

From DispersiveWiki


If is a map then a measure is invariant under if for all . 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.