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.
Speaker
Maria Saprykina, KTH Royal Institute of Technology
Abstract
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 $f:\mathcal M \equiv \mathbb T^d \times [-1,1]^d \to\mathcal M$, $f(\theta, I)=(\theta + I, I)$, and a real-analytic family of cocycles $\eta_\epsilon : \mathcal M \to \mathcal G$, indexed by a complex parameter $\epsilon$ in an open ball $\mathcal E_\rho \in\mathbb C$. 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.