A random map is a discrete time process in which one of a number of functions is selected at random and applied. Here we study random maps of
[
0
,
1
]
[0,1]
which represent dynamical systems on the square
[
0
,
1
]
×
[
0
,
1
]
[0,1] \times [0,1]
. Sufficient conditions for a random map to have an absolutely continuous invariant measure are given, and the number of ergodic components of a random map is discussed.