For $1\le r<\infty$, we construct a piecewise $C^{r}$ expanding map $F:D\to D$ on the domain $D=(0,1)\times (-1,1)\subset\mathbb{R}^{2}$ with the following property: there exists an open set $B$ in $D$ such that the diameter of $F^{n}(B)$ converges to $0$ as $n\to\infty$ and the empirical measure $n^{-1}\sum_{k=0}^{n-1}\delta_{F^{k}(x)}$ converges to the point measure $\delta_{p}$ at $p=(0,0)$ as $n \to\infty$ for any point $x\in B$.