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

Abstract

Let $T$ be the Gauss map. For each $α\in [0,1]$, let $E(α)$ be the survivor set consisting of all $x\in [0,1]$ whose orbit $\{T^k(x):k\ge 0\}$ never enters the interval $(α,1]$. We prove that the set-valued function $α\mapsto E(α)$ is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function $α\mapsto E(α)$ remains constant. We also describe the points at which the map $α\mapsto E(α)$ is not locally constant. Further, we obtain the Hausdorff dimension of $E(α)$ for all $α\in [0,1]$. We prove that the function $α\mapsto \dim_H(E(α))$ 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 $α$-continued fraction expansion and regular continued fraction expansion.