Survivor Set of Gauss Map with a Hole at $1$

Let $T$ be the Gauss map. For each $\alpha \in [0,1]$, let $E(\alpha )$ be the survivor set consisting of all $x\in [0,1]$ whose orbit $\{T^k(x):k\ge 0\}$ never enters the interval $(\alpha ,1]$. We prove that the set-valued function $\alpha \mapsto E(\alpha )$ is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function $\alpha \mapsto E(\alpha )$ remains constant. We also describe the points at which the map $\alpha \mapsto E(\alpha )$ is not locally constant. Further, we obtain the Hausdorff dimension of $E(\alpha )$ for all $\alpha \in [0,1]$. We prove that the function $\alpha \mapsto {\dim }_H(E(\alpha ))$ forms a devil’s staircase on $(0,1]$, but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical $\alpha$-continued fraction expansion and regular continued fraction expansion.