Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles

TL;DR

This paper proves that for certain higher-dimensional quasiperiodic cocycles, the top Lyapunov exponents are positive, providing uniform bounds and applications to Schrödinger operators.

Contribution

It establishes nonperturbative positivity of Lyapunov exponents for higher-dimensional quasiperiodic cocycles with large matrix blocks, extending previous results.

Findings

Top d Lyapunov exponents are bounded away from zero.

Uniform bounds on Lyapunov exponents relative to matrix block measurements.

Applications to positive lower bounds in band lattice Schrödinger operators.

Abstract

We consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schroedinger operators.