Partition zeta functions, multifractal spectra, and tapestries of complex dimensions

Abstract

For a Borel measure and a sequence of partitions on the unit interval, we define a multifractal spectrum based on coarse Holder regularity. Specifically, the coarse Holder regularity values attained by a given measure and with respect to a sequence of partitions generate a sequence of lengths (or rather, scales) which in turn define certain Dirichlet series, called the partition zeta functions. The abscissae of convergence of these functions define a multifractal spectrum whose concave envelope is the (geometric) Hausdorff multifractal spectrum which follows from a certain type of Moran construction. We discuss at some length the important special case of self-similar measures associated with weighted iterated function systems and, in particular, certain multinomial measures. Moreover, our multifractal spectrum is shown to extend to a tapestry of complex dimensions for a specific case of atomic measures. Kate E. Ellis Department of Mathematics, California State University, Stanislaus, Turlock, CA 95382 USA E-mail address: kellis1@csustan.edu Michel L. Lapidus Department of Mathematics, University of California, Riverside, CA 92521-0135 USA E-mail address: lapidus@math.ucr.edu Michael C. Mackenzie Department of Mathematics, University of Connecticut, Storrs, CT 06269 USA E-mail address: michael.mackenzie@uconn.edu John A. Rock Department of Mathematics and Statistics, California State Polytechnic University, Pomona, CA 91768 USA E-mail address: jarock@csupomona.edu • Subject class [2010]: Primary: 11M41, 28A12, 28A80. Secondary: 28A75, 28A78, 28C15 •