Nonconmutative coboundary equations over integrable systems

\def\G{\mathcal G} \def\M{\mathcal M} \def\cE{\mathcal E} We prove an analog of Livšic theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra $\G$ or a Lie group. Namely, we consider an integrable dynamical system $f:\M \equiv\torus^d \times [-1,1]^d\to \M$, $f(θ, I)=(θ+ I, I)$, and a real-analytic family of cocycles $η_\eps : \M \to \G$, indexed by a complex parameter $\eps$ in an open ball $\cE_ρ\in\CC$. We show that if $η_\eps$ has trivial periodic data, i.e., $$ η_\eps(f^{n-1}(p))\dots η_{\eps} (f(p))\cdot η_{\eps} (p)=Id $$ for each periodic point $p=f^n p$ and each $\eps \in \cE_ρ$, then there exists a real-analytic family of maps $ϕ_\eps: \M \to \G$ satisfying the coboundary equation $$ η_\eps(θ, I)=ϕ_\eps^{-1}\circ f(θ, I)\cdot ϕ_\eps (θ, I) $$ for all $(θ, I)\in \M$ and $\eps \in \cE_{ρ/2}$. We also show that if the coboundary equation above with an analytic left-hand side $η_\eps$ has a solution in the sense of formal power series in $\eps$, then it has an analytic solution.

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