Nonconmutative coboundary equations over integrable dynamics

I will present the result of a joint work with Rafael de la Llave. We prove an analog of Livshits 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 ≡T^d × [−1, 1]^d → M, f(θ, I) = (θ + I, I), and a real-analytic family of cocycles η_ϵ : M → G, indexed by a complex parameter ϵ in an open ball E_ρ ⊂ C. We show that if η_ϵ is close to identity and has trivial periodic data, i.e., η_ϵ(f^{n−1}(p)). . . η_ϵ(f(p)) · η_ϵ(p) = Id for each periodic point p = f_{n}p and each ϵ ∈ E_ρ, then there exists a real-analytic family of maps ϕ_ϵ: M → G satisfying the coboundary equation η_ϵ(θ, I) = (ϕ_ϵ ◦ f(θ, I))^{−1}· ϕϵ(θ, I) for all (θ, I) ∈ M and ϵ ∈ E_{ρ/2}. We also show that if the coboundary equation above with an analytic left-hand side ηϵ has a solution in the sense of formal power series in ϵ, then it has an analytic solution. In a work in progress with Bassam Fayad and Rafael de la Llave we show that analogous results hold for individual maps, i.e., without the use of families.