  We study families of open chaotic maps that classically share the same
asymptotic properties -- forward and backwards trapped sets, repeller
dimensions, escape rate -- but differ in their short time behavior. When these
maps are quantized we find that the fine details of the distribution of
resonances and the corresponding eigenfunctions are sensitive to the initial
shape and size of the openings. We study phase space localization of the
resonances with respect to the repeller and find strong delocalization effects
when the area of the openings is smaller than $\hbar$.
