A Herman–Avila–Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic cocycles

Abstract

A Herman–Avila–Bochi type formula is obtained for the average sum of the top$d$Lyapunov exponents over a one-parameter family of$\mathbb{G}$-cocycles, where$\mathbb{G}$is the group that leaves a certain, non-degenerate Hermitian form of signature$(c,d)$invariant. The generic example of such a group is the pseudo-unitary group$\text{U}(c,d)$or, in the case$c=d$, the Hermitian-symplectic group$\text{HSp}(2d)$which naturally appears for cocycles related to Schrödinger operators. In the case$d=1$, the formula for$\text{HSp}(2d)$cocycles reduces to the Herman–Avila–Bochi formula for$\text{SL}(2,\mathbb{R})$cocycles.