TITLE: Ruelle Perron Frobenius Operator
QUESTION [10 upvotes]: Hello everybody! Recently in my research, I came across the Perron-Frobenius operator .. I would like to intuitive interpretation of this operator, ie, physical interpretations are possible, articles (can be physical).
I wonder how this operator originated.

REPLY [9 votes]: I like to explain the Ruelle Perron Frobenius operator this way:
Suppose that $T$ is a map from some space $X$ to itself. Suppose also that $X$ has some distinguished ambient measure $\lambda$ on it (think of Lebesgue measure). One more assumption: the ambient measure is non-singular with respect to $T$ (i.e. if $A$ is a set of measure 0, then $T^{-1}A$ has measure 0 also).
Given a measure $\mu$ on $X$, the "push-forward" of $\mu$ is defined to be $\mu\circ T^{-1}$. Seems counter-intuitive, but in the case where $\mu$ is concentrated at a single point $a$ you can check that the push-forward is concentrated at $T(a)$.
The non-singularity condition implies that if $\mu$ is a measure absolutely continuous with respect to $\lambda$, then its push-forward is also absolutely continuous with respect to $\lambda$. By the Radon-Nikodym theorem, measures absolutely continuous with respect to $\lambda$ have essentially unique densities.
The Ruelle-Perron-Frobenius operator applied to $f$ gives the density of the push-forward of the measure whose density is $f$.
In more picturesque language: if $X$ is a random variable with density $f(x)$, then $T(X)$ is a random variable with density $L[f](x)$.