  We introduce pseudo-spherical non-null framed curves in the three-dimensional
anti-de Sitter spacetime and establish the existence and uniqueness of these
curves. We then give moving frames along pseudo-spherical framed curves, which
are well-defined even at singular points of the curve. These moving frames
enable us to define evolutes and focal surfaces of pseudo-spherical framed
immersions. We investigate the singularity properties of these evolutes and
focal surfaces. We then reveal that the evolute of a pseudo-spherical framed
immersion is the set of singular points of its focal surface. We also interpret
evolutes and focal surfaces as the discriminant and the secondary discriminant
sets of certain height functions, which allows us to explain evolutes and focal
surfaces as wavefronts from the viewpoint of Legendrian singularity theory.
Examples are provided to flesh out our results, and we use the hyperbolic Hopf
map to visualize these examples.
