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.



Ruelle Perron Frobenius Operator
The place to start is with the [Perron-Frobenius theorem](https://en.wikipedia.org/wiki/Perron-Frobenius_theorem), which (in its most basic form) says that a $d\times d$ matrix $A$ with only positive entries has exactly one positive eigenvector $\vec{v}$, which corresponds to the largest eigenvalue (which is real and positive). Furthermore, all the other eigenvalues are strictly smaller in absolute value, and the iterates of any non-negative vector under the matrices $A^n$ converge exponentially to the eigendirection spanned by $\vec{v}$. This can be interpreted very intuitively by looking at the action on the unit simplex $\Delta = \{\vec{w}\in \mathbb{R}^d \mid w\_i \geq 0, \|\vec{w}\|\_1 = 1\}$ given by $f(\vec{w}) = A\vec{w} / \|\vec{w}\|\_1$. This is discussed at more length in the books by [Brin and Stuck](http://books.google.com/books?id=isStwiHsoM4C&printsec=frontcover&dq=brin+stuck&hl=en&ei=1CL0TM7bJ4L6lwfq4Oi_DA&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CCwQ6wEwAA#v=onepage&q&f=false) and by [Katok and Hasselblatt](http://books.google.com/books?id=9nL7ZX8Djp4C&printsec=frontcover&dq=katok+hasselblatt&hl=en&ei=6yL0TIb-MIO0lQf2laD4DA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false).


Once you understand that case, you have at least a general picture for what you're trying to prove in the more general setting, when $\vec{w} \in \mathbb{R}^d$ is replaced by a function $\phi$ in some Banach space $X$. The idea in a wide variety of settings is that you have a dynamical system which defines a natural action on spaces of functions, and you'd like to find a Banach space of functions on which the properties of that action (which is the Ruelle Perron-Frobenius operator) mimic those of the positive matrix $A$ above. The two cases are linked by the fact that if $A$ is a $0$-$1$ matrix that determines the admissible transitions for a mixing topological Markov chain on $d$ symbols, then the matrix $A$ actually ***is*** the Perron-Frobenius operator. (Note that the mixing property says that some power of $A$ is positive, so then you can apply the standard Perron-Frobenius theorem.)


