Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators

Abstract

We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the spectrum of the corresponding Schrödinger operators is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general $\mathrm{SL}(2,\mathbb{R})$ cocycles, and obtain that uniform hyperbolic systems form a open and dense set in some one-parameter family.

21 pages, preliminary version, all comments are welcome

References in corpus (2)

• Positive Lyapunov Exponents for Quasiperiodic Szego cocycles
• Quasi-Periodic Schrödinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology

Cited by in corpus (1)

• Abundance of mode-locking for quasiperiodically forced circle maps