Two bifurcation sets arising from the beta transformation with a hole at $0$

Abstract

Given ${\hat{I}}^2\in (1,2],$ the ${\hat{I}}^2$-transformation $T_{\hat{I}}^2:x\mapsto {\hat{I}}^{2}x(\mathrm{mod}1)$ on the circle $[0,1)$ with a hole $[0,t)$ was investigated by Kalle et al.~(2019). They described the set-valued bifurcation set \[ \mathcal E_β:=\{t\in[0, 1): K_β(t')\ne K_β(t)~\forall t'>t\}, \] where $K_β(t):=\{x\in[0, 1): T_β^n(x)\ge t~\forall n\ge 0\}$ is the survivor set. In this paper we investigate the dimension bifurcation set \[ \mathcal B_β:=\{t\in[0, 1): \dim_H K_β(t')\ne \dim_H K_β(t)~\forall t'>t\}, \] where ${\dim }_H$ denotes the Hausdorff dimension. We show that if ${\hat{I}}^2\in (1,2]$ is a multinacci number then the two bifurcation sets ${\mathcal{B}}_{\hat{I}}^2$ and ${\mathcal{E}}_{\hat{I}}^2$ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for ${\hat{I}}^2$ a multinacci number we have ${\dim }_H({\mathcal{E}}_{\hat{I}}^2\cap [t,1])={\dim }_H{K}_{\hat{I}}^2(t)$ for any $t\in [0,1)$. This confirms a conjecture of Kalle et al.~for ${\hat{I}}^2$ a multinacci number.

12 pages