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Multifractal decomposition of Moran Fractals

We analyze the multifractal decomposition of Moran fractals using the Olsen Multifractal Measure (OMM). We show that this measure is non-singular when the Halsey’s partition function takes the value 1, this result has two consequences, one of them defines the mass function τ(q), the other is that it also defines a statistical measure ν in terms of a probability vector P (q). The Hausdorff dimension f(q) of the concentration set of the measure ν is equal with the Holder exponent evaluated with the measure ν, this equality implies that f(q) = q α(q) – τ(q). Finally, using the results of Cawley and Mauldin it is shown that D(α) = DimH (J α) is the Legendre Transform of τ(q), where Jα={x∈M∣limr→0ln⁡μ(B(x,r))ln⁡r=α} , i.e. D(α) = qα(q) – τ(q).