The locus of nonsimple abelian varieties in the moduli space of principally polarized abelian varieties gives rise to Noether-Lefschetz cycles. We study their intersection theoretic properties using the tautological projection constructed in [4], and show that projection defines a homomorphism when restricted to cycles supported on that locus. Using Hecke correspondences and the pullback by Torelli we prove that $[\mathcal {A}_1 \times \mathcal A_{g-1}]$ is not tautological in the sense of [38] for $g=12$ and $g\geq 16$ even. We also explore the connections between Noether-Lefschetz cycles and the Gromov-Witten theory of a moving elliptic curve.

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.