Non-ergodicity for C^1 Expanding Maps

Abstract

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.