2014/10/03 by Pedro Duarte, Silvius Klein, Duarte, Pedro +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1410.0909
openalex publication_date 2014/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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].