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

Published online by Cambridge University Press:  14 March 2014

CHRISTIAN SADEL
CHRISTIAN SADEL*
Affiliation:
University of British Columbia, 1984 Mathematics Road, Vancouver, BC, CanadaV6T 1Z2 email csadel@math.ubc.ca
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 5 , August 2015 , pp. 1582 - 1591
Copyright
© Cambridge University Press, 2014 

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