On the escape rate of unique beta-expansions

Let $1<\beta \le 2$. It is well-known that the set of points in $% [0,1/(\beta -1)]$havingunique$\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 $\beta$.