论文标题
可集成系统上的非混合耦合方程
Nonconmutative coboundary equations over integrable systems
论文作者
论文摘要
\ def \ g {\ mathcal g} \ def \ m {\ mathcal m} \ def \ ce {\ mathcal e} 我们证明了Livšic定理的类似物，用于在Banach代数$ \ g $或LIE群体中具有值的集成系统，用于实现的共同体。 也就是说，我们考虑一个可集成的动力系统$ f：\ m \ equiv \ torus^d \ times [-1,1]^d \ to \ m $，$ f（θ，i）=（θ+ i，i，i，i）$，以及一个真实的cococycles $ parive $ parive $ parions $ parions $ \ g $ \ g $， $ \ce_ρ\ in \ cc $。我们表明，如果$η_\ eps $具有微不足道的周期性数据，即$$η_\ eps（f^{n-1}（p））\点η_{\ eps}（f（p）） \ce_ρ$，然后存在一个真实的地图$ ϕ_ \ eps的家族：\ m \ to \ g $满足coboundary方程$$η_\ eps（θ，I）= ϕ__ \ eps^eps^eps^{ - 1} { - 1} \ circe f（θ，I） i）\ in \ m $和$ \ eps \ in \ ce_ {ρ/2} $。 我们还表明，如果上面具有分析性左侧$η_\ eps $具有$ \ eps $中正式功率系列的解决方案，则它具有分析解决方案。
\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.
