Consider a nice hyperbolic dynamical system $(X, T)$, for instance a $\mathcal{C}^\infty$ Anosov map. The action of the Koopman operator
$$\mathcal{K} : \ f \mapsto f \circ T$$
has a nice spectrum when acting on well-chosen Banach spaces. For instance, if $T : \mathbb{R}_{/\mathbb{Z}} \to \mathbb{R}_{/\mathbb{Z}}$ is an expanding map of the circle, one may chose to make $\mathcal{K}$ act on Sobolev spaces on negative order $H^{-m}$. The eigenvalues of $\mathcal{K}$ (Ruelle resonances) give information on the long term behaviour of sequences $(f \circ T^n)_{n \geq 0}$.
It is often useful not to work with $\mathcal{K}$, but a twisted version of $\mathcal{K}$, say
$$\mathcal{K}_\nu : \ f \mapsto e^{i \nu \varphi} \cdot f \circ T,$$
where $\varphi$ is a function of interest. This is behind a standard method to get a central limit theorem for the sequence $(\varphi \circ T^n)_{n \geq 0}$, mixing for suspension flow, and many other applications.
In Semiclassical origin of the spectral gap for transfer operators of partially expanding map, F. Faure claims that, for typical $\varphi$, the spectrum of $\mathcal{K}_\nu$ for large values of $\nu$ looks like the spectrum of a large random matrix, which he supports with a nice simulation. I have two questions (which I can split if necessary):
Question 1: Admitting that such an heuristics is reasonable, which ensemble of random matrices would best fit the operators $\mathcal{K}_\nu$? Can a somewhat precise conjecture be stated?
Question 2: Admitting that such an heuristics is reasonable, what kind of dynamical information could be deduced (on the initial system $(X,T)$, or more likely an extension as in Faure's article)?
Most ressources I could find where in the context of quantum chaos (e.g. Bohigas' Random matrix theories and chaotic dynamics) with typically the Laplacian instead of the Koopman operator and a view towards quantum mechanics, not classical dynamical systems.
