Schrödinger operators defined by continuously sampling the doubling map on the circle are expected to behave similarly to random operators, since such operators can be viewed as random operators with long-range interactions. In particular, it was conjectured that the essential spectrum consists of energy bands separated by finitely many bounded open intervals (spectral gaps). We show that there are never any spectral gaps. Along the way, we will describe some relevant background from the spectral theory of Schrödinger operators and explain the crucial ingredients for the proof, which come from topology and dynamical systems. [Joint work with D. Damanik and Í. Emilsdóttir.]