Continuity of the Lyapunov exponents for quasiperiodic cocycles

TL;DR

This paper proves the local Hölder continuity of Lyapunov spectrum blocks and the universal continuity of all Lyapunov exponents for real analytic linear cocycles over Diophantine translations, using a higher dimensional Avalanche Principle.

Contribution

It introduces a higher dimensional Avalanche Principle and demonstrates the continuity properties of Lyapunov exponents in a Banach manifold setting.

Findings

Lyapunov spectrum blocks are locally Hölder continuous.

All Lyapunov exponents are continuous everywhere.

Results extend to higher dimensional tori with a modulus of continuity loss.

Abstract

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Holder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.