The "Ten Martini Problem" dubbed after Marc Kac and Barry Simon states that the almost Mathieu operator with irrational flux has Cantor spectrum, which has been solved by Avila and Jitomirskaya completely about 10 years ago. The stronger conjecture (the dry Ten Martini Problem) predicted all spectral gaps with canonical labels are non-collapsed is still open. In this talk, I will present two equivalent formulations of the gap labeling conjecture (K-Theory and dynamical system) and discuss two corresponding approaches to tackle the dry Ten Martini problem. More precisely, I will firstly introduce the method by Choi-Elliot-Yui via writing the almost Mathieu operator as the irrational rotation C*-algebra with two canonical unitary generators. Secondly, I will introduce another approach by Avila, Avila-Jitomirskaya (with the generalization by Yuan and me), Eliasson, Sinai, and others via studying the dynamics of a family of linear skew-products driven by irrational rotation on cocycles.

I will continue the discussion from last week on the preprint https://arxiv.org/abs/2001.04383