Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators

2011/08/18 by Kai Tao, Tao, Kai · 1 citation
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.DS

paper · pdf · doi:10.48550/arxiv.1108.3747

arxiv created 2011/08/18 · openalex publication_date 2011/08/18 · arxiv updated 2011/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the quasi-periodic Jacobi operator H_x,ω in l²(ℤ) (H_x,ωϕ)(n) = -b(x+(n+1)ω)ϕ(n+1) - b(x+nω)ϕ(n-1) + a(x+nω)ϕ(n) = Eϕ(n), n∈ℤ, where a(x), b(x) are analytic function on \mathbbT, b is not identically zero, and ω obeys some strong Diophantine condition. We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent L(E) of the cocycle is positive for some E=E₀, then there exists ρ₀=ρ₀(a,b,ω,E₀), β=β(a,b,ω) such that |L(E)-L(E')|<|E-E'|^β for any E,E'∈ (E₀-ρ₀,E₀+ρ₀). If L(E)>0 for all E in some compact interval I then L(E) is Hölder continuous on I with a Hölder exponent β=β(a,b,ω,I). In our derivation we follow the refined version of the Goldstein-Schlag method \citeGS developed by Bourgain and Jitomirskaya \citeBJ.

Cited by

• Subcritical behavior for quasi-periodic Schrödinger cocycles with trigonometric potentials

Related

• Hölder continuity of the integrated density of states for quasi-periodic Jacobi operators
• Sharp Holder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles
• Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocycles
• Hölder continuity of the Lyapunov exponent for Schrödinger cocycle with quasi-periodic Gevrey potential and weak Liouville frequency
• Continuity, positivity and simplicity of the Lyapunov exponents for\n quasi-periodic cocycles