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

Abstract: For $\beta\in(1,2]$ let $T_\beta: [0,1)\to[0,1); x\mapsto \beta x\pmod 1$. 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 $1/2$ as $p\to \infty$. 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 $S_\beta(p)\to \frac{\beta^{2}+1}{\beta^4-\beta}$.