MFO Report: The dry ten Martini problem for Sturmian dynamical systems

TL;DR

This paper announces the complete solution to the Dry Ten Martini Problem for Sturmian Hamiltonians, proving all spectral gaps are open for nonzero couplings and irrational rotations, advancing understanding of aperiodic order in dynamical systems.

Contribution

It provides the first full proof of the Dry Ten Martini Problem for Sturmian systems, confirming spectral gaps are all open under specified conditions.

Findings

All spectral gaps are open for nonzero couplings.

The proof confirms predictions of the gap labeling theorem.

The result applies to all irrational rotations.

Abstract

This extended Oberwolfach report (to appear in the proceedings of the MFO Workshop 2335: Aspects of Aperiodic Order) announces the full solution to the Dry Ten Martini Problem for Sturmian Hamiltonians. Specifically, we show that all spectral gaps of Sturmian Hamiltonians (as predicted by the gap labeling theorem) are open for all nonzero couplings and all irrational rotations. We present here the proof strategy.