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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 $τ:S^1=[0,1) \to \mathbb{R}$, $ν\in \mathbb{R}$ and $φ:S^1 \to \mathbb{C}$ (a quantum state). The transformation $\hat F_ν$ acting on $φ:S^1 \to \mathbb{C}$, $\hat F_ν(φ) = ϕ$, defined by $\displaystyle ϕ(z) = \hat F_ν (φ(z)) = φ(f(z))e^{iντ(z)}$ describes a discrete time dynamical evolution of the quantum state $φ$. Given $S: \mathbb{R}\to \mathbb{R}$ we define the Lagrangian state $$φ_{x}^S(z) = \sum_{k\in\mathbb{Z}} e^{\frac{iS (z-k)}{\hbar}} e^{-\frac{(z-k-x)^2}{4\hbar}}.$$ In this case $\hat F_ν(φ_{x}^S(z)) = \sum_{k\in\mathbb{Z}}e^{\frac{iS (f(z)-k)}{\hbar}}e^{-\frac{(f(z)-k-x)^2}{4\hbar}}e^{iντ(z)}$. Under suitable conditions on $S$ the micro-support of $φ^S_x (z)$, when $\hbar \to 0$, is $(x,S'(x))$. One of meanings of the semiclassical limit in Quantum Mechanics is to take $ν=\frac{1}{\hbar}$ and $\hbar \to 0$. We address the question of finding $S$ such that $φ^S_x$ satisfies the property: $ \forall x$, we have that $\hat{F}_ν(φ^S_x)$ has micro-support on the graph of $y\to S'(y)$ (which is the micro-support of $φ^S_x$). In other words: which $S$ is such that $\hat{F}_ν$ leaves the micro-support of $φ^S_x$ invariant? This is related to a coboundary equation for $τ$, twist conditions and the boundary of the fat attractor.