I am looking for references that motivative the following definition of a transfer operator from Wikipedia:
Let $X$ be an arbitrary set and $f: X\to X$. Then the transfer operator $\mathcal{L}$ is defined on the set $\{\Phi:X\to\mathbb{C}\}$ as
$$\left(\mathcal{L}\Phi\right)(x)\sum_{y\in f^{-1}\left[\{x\}\right]}g(y)\Phi(y)$$
where $g:X\to\mathbb{C}$ is an auxiliary function.
Here by "motivating" I mean that the references in question go over examples that 1.) demonstrate why this particular definition of a transfer operator is interesting, 2.) show why other definitions would yield unsatisfactory behavior, like what if we just didn't consider the pre-image of $f$ at $x$, 3.) discuss theoretical properties of the operator.
In particular, I am interested in the points 1.) and 2.), as 3.) might be covered quite easily by some operator theoretic/functional analysis text; but still, I would be happy to learn about good references for the third point as well!
Thank you in advance!
No one source needs to cover everything, but