We will discuss continuum analogues of substitution Hamiltonians -- specifically, we will discuss Schrodinger operators on the real line whose potentials are described by an ergodic subshift over a finite alphabet and a rule that replaces symbols of the alphabet by compactly supported potential pieces. In this setting, the spectrum and the spectral type are almost surely constant, and one can identify the almost sure absolutely continuous spectrum with the Lebesgue essential closure of the set of energies with vanishing Lyapunov exponent. Using this and results of Damanik-Lenz and Klassert-Lenz-Stollmann, we can show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies a precise combinatorial condition due to Boshernitzan. We will discuss the specific case of operators over the Fibonacci subshift in detail. This is joint work with D. Damanik and A. Gorodetski.