A Modified Multifractal Formalism for a Class of Self-similar Measures with Overlap

Abstract

The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H\left\{x:\lim_{r\rightarrow 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha\right\}. \] For self-similar measures under the open set condition the behaviour of this and related functions is well-understood ([CM92],[Ols95],[AP96]); the situation turns out to be very regular and is governed by the so-called ``multifractal formalism''. Recently there has been a lot of interest in understanding how much of the theory carries over to the overlapping case; most of the results obtained, however, apply only to a limited range of $\alpha$ (the ``left half'' of the spectrum). Here we carry out a complete study of the multifractal structure for a class of self-similar measures with overlap which includes the $3$-fold convolution of the Cantor measure. Among other things, we prove that the multifractal formalism fails for many of these measures, but it holds when taking a suitable restriction.