  We study the dynamics of cold atoms subjected to {\em pairs} of closely
time-spaced $\delta$-kicks from standing waves of light. The classical phase
space of this system is partitioned into momentum cells separated by trapping
regions. In a certain range of parameters it is shown that the classical motion
is well described by a process of anomalous diffusion. We investigate in detail
the impact of the underlying classical anomalous diffusion on the quantum
dynamics with special emphasis on the phenomenon of dynamical localization.
Based on the study of the quantum density of probability, its second moment and
the return probability we identify a region of weak dynamical localization
where the quantum diffusion is still anomalous but the diffusion rate is slower
than in the classical case. Moreover we examine how other relevant time scales
such as the quantum-classical breaking time or the one related to the beginning
of full dynamical localization are modified by the classical anomalous
diffusion. Finally we discuss the relevance of our results for the
understanding of the role of classical cantori in quantum mechanics.
