2022/05/24 by Rafael de la Llave, de la Llave, Rafael, Maria Saprykina +1
Mathematics · Physics and Astronomy · #37A20 #70H08 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2205.12356
openalex publication_date 2022/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
\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 ≡\torusd × [-1,1]d→ \M, f(θ, I)=(θ+ I, I), and a real-analytic family of cocycles η_\eps : \M → \G, indexed by a complex parameter \eps in an open ball \cEρ∈\CC. We show that if η_\eps has trivial periodic data, i.e., η_\eps(fn-1(p))… η\eps (f(p))⋅ η\eps (p)=Id for each periodic point p=fn p and each \eps ∈ \cEρ, then there exists a real-analytic family of maps ϕ_\eps: \M → \G satisfying the coboundary equation η_\eps(θ, I)=ϕ_\eps-1∘ f(θ, I)⋅ ϕ_\eps (θ, I) for all (θ, I)∈ \M and \eps ∈ \cEρ/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.