Multifractal spectrum of self-similar measures with overlap

Abstract

It is well known that the multifractal spectrum of a self-similar measure satisfying the open set condition is a closed interval. Recently, there has been interest in the overlapping case and it is known that in this case there can be isolated points. We prove that for an interesting class of self-similar measures with overlap the spectrum consists of a closed interval union together with at most two isolated points. In the case of convolutions of uniform Cantor measures we determine the end points of the interval and the isolated points. We also give an example of a related self-similar measure where the spectrum is a union of two disjoint intervals. In contrast, we prove that if one considers quotient measures of this class on the quotient group [0, 1], rather than the real line, the multifractal spectrum is a closed interval.