Special Session 41: Dynamical Systems and Spectral Theory

We consider continuous SL(2, R)-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2, R)-cocycle. Using this, we show that if a cocycle’s homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be C-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrödinger operator is a Cantor set.