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

## TL;DR

This paper studies two bifurcation sets related to the beta transformation with a hole at zero, showing they coincide for multinacci numbers and providing a complete characterization, confirming a conjecture for these special beta values.

## Contribution

It characterizes the dimension bifurcation set and proves its equivalence with the set-valued bifurcation set for multinacci numbers, confirming a conjecture by Kalle et al.

## Key findings

- The two bifurcation sets coincide for multinacci numbers.
- A complete characterization of the bifurcation sets is provided.
- Confirmed the conjecture relating Hausdorff dimensions for multinacci beta values.

## Abstract

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