The $\unicode[STIX]{x1D6FD}$-transformation with a hole at 0. (September 2020) - KALLE, CHARLENE; KONG, DERONG; LANGEVELD, NIELS; LI, WENXIA
Abstract : For $\unicode[STIX]{x1D6FD}\in (1, 2]$ the $\unicode[STIX]{x1D6FD}$ -transformation $T_{\unicode[STIX]{x1D6FD}}:[0, 1)\rightarrow [0, 1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$ . For $t\in [0, 1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0, t)$ given by $$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0, 1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0, t)\text{ for all }n\geq 0\}.\end{eqnarray}$$ In this paper we characterize the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ of all parameters $t\in [0, 1)$ for which the set-valued function $t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$ is not locally constant. We show that $E_{\unicode[STIX]{x1D6FD}}$ is a Lebesgue null set of full Hausdorff dimension for all $\unicode[STIX]{x1D6FD}\in (1, 2)$ . We prove that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1, 2)$ the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\unicode[STIX]{x1D6FD}\in (1, 2)$ for which $E_{\unicode[STIX]{x1D6FD}}$ 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 $\unicode[STIX]{x1D6FD}\in (1, 2)$ a lower and an upper bound for the value $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$ such that the Hausdorff dimension of $K_{\unicode[STIX]{x1D6FD}}(t)$ is positive if and only if $t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$ . We show that $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$ for all $\unicode[STIX]{x1D6FD}\in (1, 2)$ .