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

Abstract

Given $β\in(1,2],$ the $β$-transformation $T_β: x\mapsto βx\pmod 1$ 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 $β\in(1,2]$ is a multinacci number then the two bifurcation sets $\mathcal B_β$ and $\mathcal E_β$ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for $β$ a multinacci number we have $\dim_H(\mathcal E_β\cap[t, 1])=\dim_H K_β(t)$ for any $t\in[0, 1)$. This confirms a conjecture of Kalle et al.~for $β$ a multinacci number.