Type I energies with positive Lyapunov exponent and gap-labelling condition bound open spectral gaps for irrational frequencies and trig-polynomial potentials, making the all-gaps-open property robust for perturbed almost-Mathieu operators.
and You, J., Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems,Invent
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Monotonicity, global symplectification and the stability of Dry Ten Martini Problem
Type I energies with positive Lyapunov exponent and gap-labelling condition bound open spectral gaps for irrational frequencies and trig-polynomial potentials, making the all-gaps-open property robust for perturbed almost-Mathieu operators.