Maria Saprykina: Nonconmutative coboundary equations over integrable systems

Date: 2023-03-29

Time: 11:00 - 12:00

Please note that this seminar will be held in the main building as Kuskvillan is reserved for IML on Wednesday, March 29. Zoom not available.


Maria Saprykina, KTH Royal Institute of Technology


We prove an analog of Livsic theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra $mathcal G$ (in particular, $mathcal G =GL(m,mathbb R)$) or a Lie group.  

Namely, we consider an integrable dynamical system $f:mathcal M equiv mathbb T^d times [-1,1]^d tomathcal M$, $f(theta, I)=(theta + I, I)$, and a real-analytic family of cocycles $eta_epsilon : mathcal M to mathcal G$, indexed by a complex parameter $epsilon$ in an open ball $mathcal E_rho inmathbb C$. We show that if $eta_epsilon$ has trivial periodic data, i.e., $$ eta_epsilon(f^{n-1}p)dots eta_{epsilon} (fp)cdot eta_{epsilon} (p)=Id $$ for each periodic point $p=f^n p$ and each $epsilon in mathcal E_{rho}$, then there exists a real-analytic family of maps $phi_epsilon: mathcal M to mathcal G$ satisfying the coboundary equation $$ eta_epsilon(theta, I)=phi_epsilon^{-1}circ f(theta, I)cdot phi_epsilon (theta, I) $$ for all $(theta, I)in mathcal M$ and $epsilon in mathcal E_{rho/2}$.

Joint work with Rafael de la Llave.