Multifractal Decompositions using Iterated Function Systems

Abstract

We analyze two types of multifractal decompositions (MD) of fractals F generated by an Iterated Function System (IFS), they are the geometric and the statistical MD, the first is generated by an IFS and the second by an IFS with probability. In the first, F is decomposed in subsets M(φ) of points characterized by the same vector frequency φ, and we evaluate their Hausdorff dimension (HD). In the second, F is decomposed in subsets Jα of points with the same pointwise dimension α; however Jα is composed by an infinite subsets M(φ), therefore Jα is a multifractal, this implies that its HD is the maximum HD of its components M(φ), using a maximizing procedure we find φ∗ such that HD of M(φ∗) is greater than any other M(φ) for a fixed α, this procedure gives in a natural form the auxiliary functions proposed by Cawlin and Mauldin. Thus we present a more simple description of the MD.