  Inspired by the similarity between the fractal Weierstrass function and
quantum systems with discrete scaling symmetry, we establish general conditions
under which the dynamics of a quantum system will exhibit fractal structure in
the time domain. As an example, we discuss the dynamics of the Loschmidt
amplitude and the zero-momentum occupation of a single particle moving in a
scale invariant $1/r^2$ potential. In order to show these conditions can be
realized in ultracold atomic gases, we perform numerical simulations with
practical experimental parameters, which shows that the dynamical fractal can
be observed in realistic time scales. The predication can be directly verified
in current cold atom experiments.
