  We consider the problem of finding response curves for a class of binary
two-dimensional cellular automata with $L$-shaped neighbourhood. We show that
the dependence of the density of ones after an arbitrary number of iterations,
on the initial density of ones, can be calculated for a fairly large number of
rules by considering preimage sets. We provide several examples and a summary
of all known results. We consider a special case of initial density equal to
0.5 for other rules and compute explicitly the density of ones after $n$
iterations of the rule. This analysis includes surjective rules, which in the
case of $L$-shaped neighbourhood are all found to be permutive. We conclude
with the observation that all rules for which preimage curves can be computed
explicitly are either finite or asymptotic emulators of identity or shift.
