Monotonicity, global symplectification and the stability of Dry Ten Martini Problem

TL;DR

This paper proves that for certain irrational frequencies and potentials, spectral gaps are open and stable, using a geometric approach involving symplectic cocycles and a global symplectification method.

Contribution

It introduces a novel geometric, all-frequency approach to analyze spectral gaps and their stability in the supercritical regime of the Dry Ten Martini Problem.

Findings

All spectral gaps are open for the almost-Mathieu operator in the supercritical regime.

The stability of spectral gaps persists under small perturbations at any irrational frequency.

A new geometric method involving symplectic cocycles and global symplectification is developed.

Abstract

For any fixed irrational frequency and trigonometric-polynomial potential, we show that every type I energy with positive Lyapunov exponent that satisfies the gap-labelling condition is a boundary of an open spectral gap. As a corollary, for the almost-Mathieu operator in the supercritical regime the "all spectral gaps are open" property is robust under a small trigonometric-polynomial perturbation at any irrational frequency. The proof introduces a geometric, all-frequency approach built from three ingredients: (i) the projective action on the Lagrangian Grassmannian and an associated fibred rotation number, (ii) monotonicity of one-parameter families of (Hermitian) symplectic cocycles, and (iii) a partially hyperbolic splitting with a two-dimensional center together with a global symplectification (holonomy-driven parallel transport). This provides a partial resolution to the stability of the Dry Ten Martini Problem in the supercritical regime, and answers a question by M. Shamis regarding the survival of periodic gaps.