Analytic complex one-frequency cocycles

CDSNS Colloquium
Friday, January 17, 2014 - 11:05am
1 hour (actually 50 minutes)
Skiles 005
University of British Columbia, Vancouver.
(Joint work with A. Avila and S. Jitomirskaya). An analytic, complex, one-frequency cocycle is given by a pair $(\alpha,A)$ where $A(x)$ is an analytic and 1-periodic function that maps from the torus $\mathbb(R) / \mathbb(Z)$ to the complex $d\times d$ matrices and $\alpha \in [0,1]$ is a frequency. The pair is interpreted as the map $(\alpha,A)\,:\, (x,v) \mapsto (x+\alpha), A(x) v$. Associated to the iterates of this map are (averaged) Lyapunov exponents $L_k(\alpha,A)$ and an Osceledets filtration. We prove joint-continuity in $(\alpha,A)$ of the Lyapunov exponents at irrational frequencies $\alpha$, give a criterion for domination and prove that for a dense open subset of cocycles, the Osceledets filtration comes from a dominated splitting which is an analogue to the Bochi-Viana Theorem.