Some Exceptional Phenomena in Multifractal Formalism: Part I∗

Abstract. Recently it was discovered that the 3-fold convolution of the Cantor measure μ has intricate fractal structure [HL]: the set of local dimensions μ has an isolated point and therefore the standard multifractal formalism does not hold. Our purpose here is to give a detail study of such class of examples and to understand the dilemma. This will shed some light on the multifractal structure of measures arising from iterated functions systems with overlaps, which to a large extend, is still unknown. In this Part I, we concentrate on the L-spectrum τ(q); we give a formula for τ(q) and show that it is real analytic on R except for one non-differentiable point in R−. The basic techniques are the product of matrices and the renewal theorem. In Part II, we will prove that such μ satisfies a modified multifractal formalism.