The-Hausdorff-dimension-of-the-survivor-set

Let $1<\beta <2$, the sequence $\alpha (\beta )=\alpha (\beta {)}_1\alpha (\beta {)}_2\cdots$ be the quasi-greedy $\beta$-expansion of $1$, and $t\in [0,1)$ be a bifurcation parameter. The $\beta$-transformation is defined to be $T_{\beta }(x)=\beta x(mod1)$ for $x\in [0,1)$. The Hausdorff dimension of the survivor set $K(t)=\{x\in [0,1)\text{\hspace{0.17em}⁣}:T_{\beta }^k(x)\in ̸(0,t),\mathrm{\forall }k\ge 0\}$ is equal to $-\frac{\ln \lambda }{\ln \beta }$ under the condition that ${\sum }_{i=k}^{\mathrm{\infty }}\frac{\alpha (\beta {)}_i}{{\beta }^i}\ge t$ for any $k\ge 1$, where $\lambda \in (0,1)$ is the smallest positive solution of the equation ${\sum }_{n=1}^{\mathrm{\infty }}(\alpha (\beta {)}_n-t_n)x^n=1$ with $(t_n)$ being the quasi-greedy $\beta$-expansion of $t$. And the local Hölder exponent of the Hausdorff dimension function of $K(t)$ is larger than the value of the function itself.