On the abominable properties of the Almost Mathieu operator with Liouville frequencies

We show that, for sufficiently well approximable frequencies, several spectral characteristics of the Almost Mathieu operator can be as poor as at all possible in the class of all discrete Schroedinger operators. For example, the modulus of continuity of the integrated density of states may be no better than logarithmic. Other characteristics to be discussed are homogeneity, the Parreau–Widom property, and (for the critical AMO) the Hausdorff content of the spectrum. Based on joint work with A. Avila, Y. Last, and Q. Zhou.