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

Abstract

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.