Large deviations for quasi-periodic cocycles with singularities

TL;DR

This paper establishes large deviations estimates for quasi-periodic cocycles with singularities, extending previous results to models with multiple variables and non-invertible points, and analyzing their impact on Lyapunov exponents and related structures.

Contribution

It introduces large deviations estimates for multi-variable quasi-periodic cocycles with singularities, a novel extension allowing for non-invertible points and detailed analysis of their spectral properties.

Findings

LDT estimates depend on uniform measurements of the cocycle

Continuity properties of Lyapunov exponents are established in the presence of spectral gaps

Stronger estimates and continuity results are obtained in the one-variable case

Abstract

We derive large deviations type (LDT) estimates for linear cocycles over an ergodic multifrequency torus translation. These models are called quasi-periodic cocycles. We make the following assumptions on the model: the translation vector satisfies a generic Diophantine condition, and the fiber action is given by a matrix valued analytic function of several variables which is not identically singular. The LDT estimates obtained here depend on some uniform measurements on the cocycle. Our general results derived in [9] regarding the continuity properties of the Lyapunov exponents (LE) and of the Oseledets filtration and decompositions are then applicable, and we obtain local weak-Holder continuity of these quantities in the presence of gaps in the Lyapunov spectrum. The main new feature of this work is allowing a cocycle depending on several variables to have singularities, i.e. points of non invertibility. This requires a careful analysis of the set of zeros of certain analytic functions of several variables and of the singularities (i.e. negative infinity values) of pluri-subharmonic functions related to the iterates of the cocycle. A refinement of this method in the one variable case leads to a stronger LDT estimate and in turn to a stronger, nearly-Holder modulus of continuity of the LE, Oseledets filtration and Oseledets decomposition. This is a draft of a chapter in our forthcoming research monograph [9].