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Abstract: For a Borel measure and a sequence of partitions on the unit interval, wedefine a multifractal spectrum based on coarse Holder regularity. Specifically,the coarse Holder regularity values attained by a given measure and withrespect to a sequence of partitions generate a sequence of lengths or rather,scales which in turn define certain Dirichlet series, called the partitionzeta functions. The abscissae of convergence of these functions define amultifractal spectrum whose concave envelope is the geometric Hausdorffmultifractal spectrum which follows from a certain type of Moran construction.We discuss at some length the important special case of self-similar measuresassociated with weighted iterated function systems and, in particular, certainmultinomial measures. Moreover, our multifractal spectrum is shown to extend toa tapestry of complex dimensions for two specific atomic measures.



Author: Kate E. Ellis, Michel L. Lapidus, Michael C. Mackenzie, John A. Rock

Source: https://arxiv.org/



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