Nonconmutative coboundary equations over integrable systems

TL;DR

This paper extends Livšic's theorem to real-analytic cocycles over integrable systems with values in Banach algebras or Lie groups, establishing conditions for the existence of analytic solutions to coboundary equations.

Contribution

It proves an analog of Livšic's theorem for real-analytic cocycles over integrable systems with values in Banach algebras or Lie groups, including conditions for solutions.

Findings

Trivial periodic data implies existence of a real-analytic coboundary.

Analytic solutions exist if formal power series solutions are present.

Results apply to cocycles over a specific integrable system.

Abstract

\def\G{\mathcal G} \def\M{\mathcal M} \def\cE{\mathcal E} We prove an analog of Liv\v{s}ic 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 $f:\M \equiv\torus^d \times [-1,1]^d\to \M$, $f(\theta, I)=(\theta + I, I)$, and a real-analytic family of cocycles $\eta_\eps : \M \to \G$, indexed by a complex parameter $\eps$ in an open ball $\cE_\rho \in\CC$. We show that if $\eta_\eps$ has trivial periodic data, i.e., $$ \eta_\eps(f^{n-1}(p))\dots \eta_{\eps} (f(p))\cdot \eta_{\eps} (p)=Id $$ for each periodic point $p=f^n p$ and each $\eps \in \cE_{\rho}$, then there exists a real-analytic family of maps $\phi_\eps: \M \to \G$ satisfying the coboundary equation $$ \eta_\eps(\theta, I)=\phi_\eps^{-1}\circ f(\theta, I)\cdot \phi_\eps (\theta, I) $$ for all $(\theta, I)\in \M$ and $\eps \in \cE_{\rho/2}$. We also show that if the coboundary equation above with an analytic left-hand side $\eta_\eps$ has a solution in the sense of formal power series in $\eps$, then it has an analytic solution.