Multiplicative Jensen’s formula and quantitative global theory of one-frequency Schrödinger operators

We introduce the concept of dual Lyapunov exponents, leading to a multiplicative version of the classical Jensen’s formula for one-frequency analytic Schrödinger cocycles. This formula, in particular, gives a new proof and a quantitative version of the fundamentals of Avila’s global theory [3], fully explaining the behavior of complexified Lyapunov exponent through the dynamics of the dual cocycle. The key concepts of (sub/super) critical regimes and acceleration are all explained (in a quantitative way) through the duality approach. In particular, for trigonometric polynomial potentials, we establish partial hyperbolicity of the dual symplectic cocycle and show that the acceleration is equal to half the dimension of its center, this holding also in the appropriate sense for the general analytic case. These results lead to a number of powerful spectral and physics applications.