We consider the spectral gaps of quasi-periodic Schrodinger operators with Liouville frequencies. By establishing quantitative reducibility of the associated Schrodinger cocycle, we show that the size of the spectral gaps decays exponentially. This is a joint work with Wencai Liu.
Toeplitz sequences are constructed from periodic sequences with undetermined positions by successively filling these positions with the letters of other periodic sequences. In this talk we will consider the class of so called simple Toeplitz sequences. We will describe combinatorial properties, such as the word complexity, of the subshifts that are associated with them. The relation between combinatorial properties of the coding sequences and the Boshernitzan condition will be also discussed.
We study the spectral transition line of the extended Harper's model
in the positive Lyapunov exponent regime. We show that both pure point
spectrum and purely singular continuous spectrum occur for dense subsets
of frequencies on the transition line.