Given a random sequence of holomorphic maps
f
1
,
f
2
,
f
3
,
…
f_1,f_2,f_3,\ldots
of the unit disk
Δ
\Delta
to a subdomain
X
X
, we consider the compositions
\[
F
n
=
f
1
∘
f
2
∘
…
f
n
−
1
∘
f
n
.
F_n=f_1 \circ f_{2} \circ \ldots f_{n-1} \circ f_n.
\]
The sequence
{
F
n
}
\{F_n\}
is called the iterated function system coming from the sequence
f
1
,
f
2
,
f
3
,
…
.
f_1,f_2,f_3,\ldots .
We prove that a sufficient condition on the domain
X
X
for all limit functions of any
{
F
n
}
\{F_n\}
to be constant is also necessary. We prove that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.