Semiclassical limits, Lagrangian states and coboundary equations

Assume that f is a continuous transformation f:S¹ → S¹. We consider here the cases where f is either the transformation f(z)=z² or f is a smooth diffeomorphism of the circle S¹. Consider a fixed continuous potential τ:S¹=[0,1) → mathbb{R}, νin mathbb{R} and varphi:S¹ → mathbb{C} (a quantum state). The transformation hat F_ν acting on varphi:S¹ → mathbb{C}, hat F_ν\(varphi\) = φ, defined by displaystyle φ(z) = hat F_ν \(varphi(z\)) = varphi(f(z))e^iντ(z) describes a discrete time dynamical evolution of the quantum state varphi. Given S: mathbb{R}→ mathbb{R} we define the Lagrangian state $varphi_x^S(z) = sum_{kinmathbb{Z}} e^{frac{iS (z-k)}{hbar}} e^{-frac{(z-k-x)²}{4hbar}}.In this case\hat F_νvarphi_x^S(z) = \sum_{k\in\mathbb{Z}}e^{\frac{iS (f(z)-k)}{\hbar}}e^{-\frac{(f(z)-k-x)^2}{4\hbar}}e^{iντ(z)}. Under suitable conditions onSthe micro-support of\varphi^S_x (z), when\hbar \to 0, is(x,S'(x)). One of meanings of the semiclassical limit in Quantum Mechanics is to takeν=\frac{1}{\hbar}and\hbar \to 0. We address the question of findingSsuch that\varphi^S_xsatisfies the property: \forall x, we have that\hat{F}_νvarphi^S_xhas micro-support on the graph ofy\to S'(y)\(which is the micro-support of\varphi^S_x\). In other words: whichSis such that\hat{F}_νleaves the micro-support of\varphi^S_xinvariant? This is related to a coboundary equation forτ$, twist conditions and the boundary of the fat attractor.