Multifractal analysis of Bernoulli measures on a class of homogeneous Cantor sets

We study multifractal properties of Bernoulli measure μ_p supported on homogeneous Cantor set determined by ([0,1],(2)_k≥1,(c_k)_k≥1), where 0<p<1/2 and c_k is either d₁ or d₂ with 0<d₁<d₂<1/2. Let θ_ and θ‾ be the lower and upper limits of the occurrence frequency of d₁ in (c_k)_k≥1, respectively. It is proved by Wu (2005) [14] that, if θ_=θ‾ then the multifractal formalism holds. After given some conditions on (c_k)_k≥1, we get multifractal spectra for any 0≤θ_<θ‾≤1. We show that if θ_=0 and θ‾=1, the refined formalism does not hold, while Olsen's concave multifractal formalism holds in some cases. This answers a question asked by Olsen (1995) [10]. We also show that if 0<θ_<θ‾<1, Olsen's concave multifractal formalism does not hold.