Quantitative global theory of one-frequency quasiperiodic operators

We explore Avila's global theory for one-frequency Schrödinger operators from the point of view of Aubry duality. We establish the one-to-one correspondence between the Lyapunov exponent of the Schrödinger cocycles $(\alpha,S_E^{V}(\cdot+i\epsilon))$ and the Lyapunov exponents of its dual cocycles. We also give a new equivalent definition of subcritical, critical, supercritical regimes. Especially, we prove that the acceleration of the Schrödinger cocycle equals to the pair of zero Lyapunov exponents of its dual cocycle. This is a joint work with Svetlana Jitomirskaya, Jiangong You and Qi Zhou.