We consider aperiodic shifts of finite type σ with an equilibrium state m and associated skew-products σf where f: X→G is Hölder and G is a compact Lie group. We show that generically σf is weak-mixing and give constructive methods for achieving weak-mixing by perturbing an arbitrary f at a finite number of small neighbourhoods. When σf is not ergodic we describe the (closed) ergodic decomposition precisely. The key result shows that certain measurable eigenfunctions are essentially Hölder continuous. This leads to conditions for weak-mixing or ergodicity in terms of a functional equation which involves Hölder rather than measurable functions.