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

TL;DR

This paper extends the Herman-Avila-Bochi formula to higher-dimensional cocycles associated with pseudo-unitary and hermitian-symplectic groups, providing a new tool for analyzing Lyapunov exponents in these settings.

Contribution

The authors derive a Herman-Avila-Bochi type formula for the average sum of top Lyapunov exponents for higher-dimensional G-cocycles, generalizing known results for SL(2,R).

Findings

Derived a new formula for Lyapunov exponents of pseudo-unitary cocycles.

Connected the formula to cocycles related to Schrödinger operators.

Reduced to known formulas in special cases like d=1.

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 G-cocycles, where 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 U(c,d) or in the case c=d, the hermitian-symplectic group HSp(2d) which naturally appears for cocycles related to Schr\"odinger operators. In the case d=1, the formula for HSp(2d) cocycles reduces to the Herman-Avila-Bochi formula for SL(2,R) cocycles.