Discontinuity of Lyapunov exponent in spaces of quasiperiodic cocycles: Smoothness vs Arithmetic

2025/10/28 by Liang, Jinhao, Tao, Kai, You, Jiangong
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.24041

Abstract

We construct examples of discontinuity of Lyapunov exponent in the spaces of quasiperiodic SL(2,\mathbb R)-cocycles for fixed irrational frequencies. Especially, we prove that the Gevrey space G² is the transition space of continuity for all strong Diophantine frequencies. We also construct examples of discontinuity for other frequencies in less smooth spaces, which show that the more difficult it is to approximate the frequency with rational numbers, the more likely it is to exhibit discontinuity in smoother spaces.

Citations

• On the abominable properties of the almost Mathieu operator with well approximated frequencies
• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles
• 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 C² Quasiperiodic Schrödinger Cocycles 
• Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential
• Holder Continuity of the Integrated Density of States for Quasi-Periodic Schrodinger Equations and Averages of Shifts of Subharmonic Functions

Related

• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles
• Continuity of the Lyapunov exponent for analytic multi-frequency quasiperiodic cocycles
• Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles
• Discontinuity of Lyapunov exponents for SL(2,ℝ) valued cocycles
• Quasi-Periodic Schrödinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology