  The correlation between level velocities and eigenfunction intensities
provides a new way of exploring phase space localization in quantized
non-integrable systems. It can also serve as a measure of deviations from
ergodicity due to quantum effects for typical observables. This paper relies on
two well known paradigms of quantum chaos, the bakers map and the standard map,
to study correlations in simple, yet chaotic, dynamical systems. The behaviors
are dominated by the presence of several classical structures. These primarily
include short periodic orbits and their homoclinic excursions. The dependences
of the correlations deriving from perturbations allow for eigenfunction
features violating ergodicity to be selectively highlighted. A semiclassical
theory based on periodic orbit sums leads to certain classical correlations
that are super-exponentially cut off beyond a logarithmic time scale. The
theory is seen to be quite successful in reproducing many of the quantum
localization features.
