2004/02/20 by Raphaël Krikorian, Krikorian, Raphaël
Materials Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Magnetism in coordination complexes #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.DS
paper · pdf · doi:10.48550/arxiv.math/0402333
80 pages
arxiv created 2004/02/20 · openalex publication_date 2004/02/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given α in some set Σ of total (Haar) measure in \bf T=\bf R/\bf Z, and A∈ C∞(\bf T,SL(2,\bf R)) which is homotopic to the identity, we prove that if the fibered rotation number of the skew-product system (α,A):\bf T× SL(2,\bf R)→ \bf T× SL(2,\bf R), (α,A)(θ,y)=(θ+α,A(θ)y) is diophantine with respect to α and if the fibered products are uniformly bounded in the C0-topology then the cocycle (α,A) is C^∞-reducible --that is A(⋅)=B(⋅+α)A0 B(⋅)-1, for some A0∈ SL(2,\bf R), B∈ C∞(\bf T,SL(2,\bf R)). This result which can be seen as a non-pertubative version of a theorem by L.H. Eliasson has two interesting corollaries: the first one is a result of differentiable rigidity: if α∈Σ and the cocycle (α,A) is C0-conjugated to a constant cocycle (α,A0) with A0 in a set of total measure in SL(2,\bf R) then the conjugacy is C^∞; the second consequence is: if α∈ Σ is fixed then the set of A∈ C^∞(\bf T,SL(2,\bf R)) for which (α,A) has positive Lyapunov exponent is C^∞-dense. A similar result is true for the Schrödinger cocycle and for 2-frequencies conservative differential equations in the plane.