A dynamical Thouless formula

2022/08/11 by Jamerson Bezerra, Ao Cai, Bezerra, Jamerson +7 · 2 citations
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2208.06022

openalex publication_date 2022/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

In this paper we establish an abstract, dynamical Thouless-type formula for affine families of GL (2,ℝ) cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schrödinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive Hölder-type continuity properties of the fibered rotation number for linear cocycles over various base dynamics.

Cited by

• The fibered rotation number
• Obstructions to the regularity of the Lyapunov exponents for non-compact random Schrödinger cocycles

Related

• The fibered rotation number
• The fibered rotation number for ergodic symplectic cocycles and its applications: I. Gap Labelling Theorem
• Log Hölder continuity of the rotation number
• KAM, Lyapunov exponents, and the Spectral Dichotomy for typical one-frequency Schrodinger operators
• Lyapunov exponents for families of rotated linear cocycles