Invariant measures: Difference between revisions
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If <math>\phi</math> is a map then a measure <math>\mu</math> is '''invariant''' under <math>\phi</math> if <math>\mu(\phi^{-1}A) = \mu(A)</math> for all <math>A</math>. 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 == | == Examples of invariant measures == | ||
Latest revision as of 19:11, 8 January 2007
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.
