It is shown that for any irrational rotation number and any admissible gap labelling number the almost Mathieu operator (also known as Harper’s operator) has a gap in its spectrum with that labelling number. This answers the strong version of the so-called "Ten Martini Problem". When specialized to the particular case where the coupling constant is equal to one, it follows that the "Hofstadter butterfly" has for any quantum Hall conductance the exact number of components prescribed by the recursive scheme to build this fractal structure.
"Persistence of gaps in the spectrum of certain almost periodic operators." Adv. Theor. Math. Phys. 16 (2) 693 - 712, April 2012.