The TMP was fully solved by Artur Avila and Svetlana Jitomirskaya in 2005. The Dry TMP has been recently announced to be solved for the non-critical case (coupling constant different than one). The corresponding preprint by Artur Avila, Jiangong You and Qi Zhou appeared on June 2023.

This talk is about the Dry TMP for Sturmian Hamiltonians. These are one-dimensional Schroedinger operators with aperiodic potentials determined by Sturmian sequences. The potential is determined in terms of two parameters: the frequency and the potential strength (a.k.a coupling constant). As for the Almost-Mathieu operator the Dry TMP is whether all the possible spectral gaps are there for all irrational frequencies and all coupling constants. For large values of the coupling constant, the Sturmian Dry TMP was solved by Raymond in 1997. In 2016, David Damanik, Anton Gorodetski and William Yessen provided a solution if the frequency is the golden mean and for all couplings.

In a current project with Siegfried Beckus and Raphael Loewy we solve the Sturmian Dry TMP for all irrational frequencies and all couplings. In the talk we present the problem and the route to its resolution.