Continuity of the Lyapunov exponents for quasiperiodic cocycles

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.

63 pages, 1 figure

References in corpus (5)

• Complex one-frequency cocycles
• Localization for quasiperiodic Schrodinger operators with multivariable Gevrey potential functions
• Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles
• Continuity of Lyapunov Exponents for Random 2D Matrices
• Regularity and convergence rates for the Lyapunov exponents of linear co-cycles

Cited by in corpus (11)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Complex one-frequency cocycles
• Dominated splittings and the spectrum of singular quasi-periodic Jacobi operators
• Large deviation estimates and Hölder regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles
• Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schrödinger Cocycles 
• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles
• An abstract continuity theorem of the Lyapunov exponents
• Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ${\ell }^2({\mathbb{Z}}^d)$
• The Avalanche Principle and other estimates on Grassmann manifolds
• Large deviations for quasi-periodic cocycles with singularities
• Anderson localization for one-frequency quasi-periodic block operators with long-range interactions