Continuity of the Lyapunov exponent for analytic multi-frequency quasiperiodic cocycles

TL;DR

This paper proves the continuity of the Lyapunov exponent for analytic multi-frequency quasiperiodic cocycles, extending known results from single-frequency cases to more complex multi-frequency systems, with applications to Jacobi cocycles.

Contribution

It extends Bourgain's continuity result of the Lyapunov exponent from single-frequency to multi-frequency quasiperiodic cocycles in the matrix setting.

Findings

Lyapunov exponent is continuous for multi-frequency cocycles.

Results apply to multifrequency Jacobi cocycles with periodic backgrounds.

Generalizes known single-frequency continuity results to multi-frequency cases.

Abstract

It is known that the Lyapunov exponent of analytic 1-frequency quasiperiodic cocycles is continuous in cocycle and, when the frequency is irrational, jointly in cocycle and frequency. In this paper, we extend a result of Bourgain to show the same continuity result for multifrequency quasiperiodic $M(2,\mathbb{C})$ cocycles. Our corollaries include applications to multifrequency Jacobi cocycles with periodic background potentials.