  Let $1<\beta \leq 2$. It is well-known that the set of points in $%
[0,1/(\beta -1)]$ having unique $\beta $-expansion, in other words, those
points whose orbits under greedy $\beta $-transformation escape a hole
depending on $\beta $, is of zero Lebesgue measure. The corresponding escape
rate is investigated in this paper. A formula which links the Hausdorff
dimension of univoque set and escape rate is established in this study. Then we
also proved that such rate forms a devil's staircase function with respect to
$\beta $.
