Periodic points in the $\beta$-transformation with a hole at 0

For $\beta \in (1,2]$ let $T_{\beta }:[0,1)\to [0,1);x\mapsto \beta x(\mathrm{m}\mathrm{o}\mathrm{d}1)$. In this paper we study the periodic points in the open dynamical system $([0,1),T_{\beta })$ with a hole $[0,t)$. For $p\in \mathbb{N}$ we characterize the largest $t$, denoted by $S_{\beta }(p)$, in which the survivor set $K_{\beta }(t)$ has a periodic point of smallest period $p$. More precisely, we give precise formulae for this critical value $S_{\beta }(p)$ when $\beta =2$, $\beta =\frac{1+\sqrt{5}}{2}$ and $\beta$ being the tribonacci number. We show that for $\beta =2$ the critical value $S_2(p)$ converges to $1\mathrm{/}2$ as $p\to \mathrm{\infty }$. When $\beta =\frac{1+\sqrt{5}}{2}$, the critical value $S_{\beta }(p)\to \frac{1}{{\beta }^3-\beta }$. While $\beta$ is the tribinacci number, the critical value $S_{\beta }(p)\to \frac{{\beta }^2+1}{{\beta }^4-\beta }$.