Given β∈(1,2], the β-transformation Tβ:x↦βx(mod1) on the circle [0,1) with a hole [0,t) was investigated by Kalle
et al.~(2019). They described the set-valued bifurcation set
Eβ:={t∈[0,1):Kβ(t′)=Kβ(t)∀t′>t},
where Kβ(t):={x∈[0,1):Tβn(x)≥t∀n≥0} is the
survivor set. In this paper we investigate the dimension bifurcation set
Bβ:={t∈[0,1):dimHKβ(t′)=dimHKβ(t)∀t′>t},
where dimH denotes the Hausdorff dimension.
We show that if β∈(1,2] is a multinacci number then the two
bifurcation sets Bβ and Eβ 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
dimH(Eβ∩[t,1])=dimHKβ(t) for any t∈[0,1).
This confirms a conjecture of Kalle et al.~for β a multinacci number.Comment: 12 page