We consider the one-dimensional Schrodinger operator with potential given by Period Doubling and Thue-Morse sequences. We describe the results of Bellisard et al. in which the spectrum of these operators are obtained through the trace map associated to this operator. We see that the spectrum for non-zero values of the potential is a Cantor set of zero Lebesque measure, and give explicit description of the spectral gaps.

The graph G_0 was introduced by Kechris-Solecki-Todorcevic in the late 90s,
and has since turned into an essential object in descriptive set theory. In
joint work with Richard Ketchersid, we prove a version of the G_0-dichotomy
in models of AD^+. This is then used to establish that the quotient by the
equivalence relation E_0 is a successor of R, a result previously known
under AD_R, but (perhaps surprisingly) not in L(R).