  We show that the position operator for a class of $f$-deformed oscillators
has a fractal spectrum, homeomorphic to the Cantor set, via a unitary
transformation to Harper's model. The class corresponds to a choice of ergodic
operators for the deformation function. Hofstadter's butterfly is plotted by
direct diagonalization of a position operator with an originally vanishing
diagonal. This is equivalent to a one-dimensional hamiltonian without
potential.
