Perron-Frobenius Operator

An Operator which describes the time evolution of densities in Phase Space. The Operator can be defined by

$$\rho_{n+1}=\tilde L \rho_n,$$

where $\rho_n$ are the Natural Densities after the $n$th iteration of a map $f$. This can be explicitly written as

$$\tilde L\rho(y)=\sum_{x\in f^{-1}(y)} {\rho(x)\over \vert f'(x)\vert}.$$

References

Beck, C. and Schlögl, F. ``Transfer Operator Methods.'' Ch. 17 in Thermodynamics of Chaotic Systems. Cambridge, England: Cambridge University Press, pp. 190-203, 1995.