A formula with some applications to the theory of Lyapunov exponents

Abstract

We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. Indeed, we show that equality holds in Herman's result. Finally, we give a result about the growth of the spectral radius of products.

11 pages, no figures, final version, accepted by the Israel Journal of Mathematics

References in corpus (1)

• Genericity of zero Lyapunov exponents

Cited by in corpus (16)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Lyapunov exponents for random perturbations of some area-preserving maps including the standard map
• How chaotic are strange nonchaotic attractors
• Positive Lyapunov Exponents for Quasiperiodic Szego cocycles
• Non-conformal repellers and the continuity of pressure for matrix cocycles
• The generalized Berger-Wang formula and the spectral radius of linear cocycles
• Generic singular continuous spectrum for ergodic Schrödinger operators
• Density of positive Lyapunov exponents for SL(2,R) cocycles
• Density of positive Lyapunov exponents for symplectic cocycles
• A new proof of the Herman-Avila-Bochi formula for Lyapunov exponents of SL(2,R)-cocycles
• On $SL(2,\mathbb{R})$-cocycles over irrational rotations with secondary collisions
• Some connections between almost periodic and periodic discrete Schroedinger operators with analytic potentials
• Finer filtration for matrix-valued cocycle based on Oseledec's multiplicative ergodic theorem
• Anderson localization for electric quantum walks and skew-shift CMV matrices
• Asymptotic behavior of the spectral radius of locally constant strongly irreducible cocycles
• Formulas for Lyapunov exponents