  Random dynamical systems (RDS) evolve by a dynamical rule chosen
independently with a certain probability, from a given set of deterministic
rules. These dynamical systems in an interval reach a steady state with a
unique well-defined probability density only under certain conditions, namely
Pelikan's criterion. We investigate and characterize the steady state of a
bounded RDS when Pelikan's criterion breaks down. In this regime, the system is
attracted to a common fixed point (CFP) of all the maps, which is attractive
for at least one of the constituent mapping functions. If there are many such
fixed points, the initial density is shared among the CFPs; we provide a
mapping of this problem with the well known hitting problem of random walks and
find the relative weights at different CFPs. The weights depend upon the
initial distribution.
