The $β$-transformation with a hole at 0
For $β\in(1,2]$ the $β$-transformation $T_β: [0,1) \to [0,1)$ is defined by $T_β( x) = βx \pmod 1$. For $t\in[0, 1)$ let $K_β(t)$ be the survivor set of $T_β$ with hole $(0,t)$ given by \[K_β(t):=\{x\in[0, 1): T_β^n(x)\not \in (0, t) \textrm{ for all }n\ge 0\}.\] In this paper we characterise the bifurcation set $E_β$ of all parameters $t\in[0,1)$ for which the set valued function $t\mapsto K_β(t)$ is not locally constant. We show that $E_β$ is a Lebesgue null set of full Hausdorff dimension for all $β\in(1,2)$. We prove that for Lebesgue almost every $β\in(1,2)$ the bifurcation set $E_β$ contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of $β\in(1,2)$ for which $E_β$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_2$, the bifurcation set of the doubling map. Finally, we give for each $β\in (1,2)$ a lower and upper bound for the value $τ_β$, such that the Hausdorff dimension of $K_β(t)$ is positive if and only if $t< τ_β$. We show that $τ_β\le 1-\frac1β$ for all $β\in (1,2)$.