  In this paper, we consider the question of existence and uniqueness of
absolutely continuous invariant measures for expanding $C^1$ maps of the
circle. This is a question which arises naturally from results which are known
in the case of expanding $C^k$ maps of the circle where $k\geq 2$, or even
$C^{1+\epsilon}$ expanding maps of the circle. In these cases, it is known that
there exists a unique absolutely continuous invariant probability measure by
the so-called `Folklore Theorem'. It follows that this measure is ergodic. It
has been shown however that for $C^1$ maps there need not be any such measure.
However, this leaves the question of whether there can be more than one such
measure for $C^1$ expanding maps of the circle. This is the subject of this
paper, and in it, we show that there exists a $C^1$ expanding map of the circle
which has more than one absolutely continuous invariant probability measure.
