A Modified Multifractal Formalism for a Class of Self-Similar Measures

Abstract

The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H {x:\lim_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha}. \] For self-similar measures under the open set condition the behavior of this and related functions is well-understood; 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; however, much less is known in this case and what is known makes it clear that more complicated phenomena are possible. 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.