Transfer operator

Summary

The transfer operator of a map $f$ with weight $g$ acts on functions $\varphi$. The value of the new function at $x$ is the sum of $g \varphi$ over the preimages of $x$. It moves densities forward under the map, while the Koopman operator moves observables, and the two are dual to each other. Invariant densities are fixed points of the transfer operator, and its spectrum controls the speed of mixing and decay of correlations. It is also called the Ruelle operator or, for the weight given by the Jacobian, the Perron–Frobenius operator.