Semiclassical limits, Lagrangian states and coboundary equations

Assume that $f$ is a continuous transformation $f:S^1\to S^1$. We consider here the cases where $f$ is either the transformation $f(z)=z^2$ or $f$ is a smooth diffeomorphism of the circle $S^1$. Consider a fixed continuous potential $\tau :S^1=[0,1)\to \mathbb{R}$, $\nu \in \mathbb{R}$ and $\varphi:S^1 \to $\mathbb{C}$$(aquantumstate)\mathrm{.}Thetransformation$\hat F_{\nu}$actingon$\varphi:S^1 \to $\mathbb{C}$$,$\hat F_{\nu}(\varphi) = \phi$,definedby$\displaystyle \phi(z) = \hat F_{\nu} (\varphi(z)) = \varphi(f(z))e^{i\nu\tau(z)}$ describes a discrete time dynamical evolution of the quantum state $\phi$. Given $S:\mathbb{R}\to \mathbb{R}$ we define the Lagrangian state $${\phi }_x^S(z)=\sum _{k\in \mathbb{Z}}{e}^{\frac{iS(z-k)}{\mathrm{\hslash }}}{e}^{-\frac{(z-k-x{)}^2}{4\mathrm{\hslash }}}\mathrm{.}$$ In this case $\hat F_{\nu}(\varphi_{x}^S(z)) = \sum_{k\in$\mathbb{Z}$}e^{\frac{iS (f(z)-k)}{\hbar}}e^{-\frac{(f(z)-k-x)^2}{4\hbar}}e^{i\nu\tau(z)}$. Under suitable conditions on $S$ the micro-support of ${\phi }_x^S(z)$, when $\hbar \to 0$,is$(x,S’(x))$. One of meanings of the semiclassical limit in Quantum Mechanics is to take $\nu =\frac{1}{\mathrm{\hslash }}$ and $\mathrm{\hslash }\to 0$. We address the question of finding $S$ such that ${\phi }_x^S$ satisfies the property: $\mathrm{\forall }x$, we have that ${\widehat{F}}_{\nu }({\phi }_x^S)$ has micro-support on the graph of $y\to S^{\mathrm{\prime }}(y)$ (which is the micro-support of ${\phi }_x^S$). In other words: which $S$ is such that ${\widehat{F}}_{\nu }$ leaves the micro-support of ${\phi }_x^S$ invariant? This is related to a coboundary equation for $\tau$, twist conditions and the boundary of the fat attractor.