  We address the issue of fluctuations, about an exponential lineshape, in a
pair of one-dimensional kicked quantum systems exhibiting dynamical
localization. An exact renormalization scheme establishes the fractal character
of the fluctuations and provides a new method to compute the localization
length in terms of the fluctuations. In the case of a linear rotor, the
fluctuations are independent of the kicking parameter $k$ and exhibit
self-similarity for certain values of the quasienergy. For given $k$, the
asymptotic localization length is a good characteristic of the localized
lineshapes for all quasienergies. This is in stark contrast to the quadratic
rotor, where the fluctuations depend upon the strength of the kicking and
exhibit local "resonances". These resonances result in strong deviations of the
localization length from the asymptotic value. The consequences are
particularly pronounced when considering the time evolution of a packet made up
of several quasienergy states.
