Second-Order Self-Similar Identities and Multifractal Decompositions

Motivated by the study of convolutions of the Cantor measure, we set up a framework for computing the multifractal L ^q-spectrum τ(q), q > 0, for certain overlapping self-similar measures which satisfy a family of second-order identities introduced by Strichartz et al. We apply our results to the family of iterated function systems Sjx = (1/m)x + [(m - 1)m]/j, j = 0, 1, ..., m, where m is an odd integer, and obtain closed formulas defining τ(q), q > 0, for the associated self-similar measures. As a result, we can show that τ(q) is differentiable on (0, ∞) and justify the multifractal formalism in the region q > 0. Furthermore, expressions for the Hausdorff and entropy dimensions of these measures can also be derived. By letting m = 3, we obtain all these results for the 3-fold convolution of the standard Cantor measure.