  We revisit processes generated by iterated random functions driven by a
stationary and ergodic sequence. Such a process is called strongly stable if a
random initialization exists, for which the process is stationary and ergodic,
and for any other initialization, the difference between the two processes
converges to zero almost surely. Under some mild conditions on the
corresponding recursive map, without any condition on the driving sequence, we
show the strong stability of iterations. Several applications are surveyed such
as generalized autoregression and queuing. Furthermore, new results are deduced
for Langevin-type iterations with dependent noise and for multitype branching
processes.
