Multifractal analysis for convolutions of overlapping Cantor measures (Q549105)

The paper focuses on the evaluation of the local dimension of the convolution of Cantor-type measures. The local dimension of a measure \(\mu\) at a point \(x\) in its support is defined by \[ \text{dim}_{loc}\mu(x)=\lim_{r\rightarrow 0^+}\frac{\log \mu(B(x,r))}{\log r}, \] and it has its variations using \(\liminf\) or \(\limsup\). The set of all values that appear as local dimension of a measure at various points forms the multifractal spectrum. The authors consider general Cantor sets, where the length of the removed intervals is the same for all the intervals at a given step, but can vary with each step. The Cantor measures \(\mu\) are allowed to be non-uniform, so the probabilities associated to different pieces of the Cantor set are allowed to be different. Then these Cantor measures are convoluted \(m\) times, the result being denoted by \(\mu^m\). Particular attention is paid for the the local dimension of \(\mu^m\) at the lower and upper ends of the support. These numbers are evaluated and it is shown that for large values of \(m\), these numbers are isolated in the multifractal spectrum. The gap between these numbers and the rest of the multifractal spectrum is shown to increase with \(m\). The second part of the paper is concerned with the 3-fold convolution of the uniform Cantor measure on the usual Cantor set \(C(1/3)\) considered now as a subset of the torus, i.e., addition \(\mod 1\). In this case the multifractal spectrum is completely described.