Convolutions of equicontractive self-similar measures on the line

Abstract

Let $\mu$ be a self-similar measure on $\mathbb{R}$ generated by an equicontractive iterated function system. We prove that the Hausdorff dimension of $\mu^{*n}$ tends to $1$ as $n$ tends to infinity, where $\mu^{*n}$ denotes the $n$-fold convolution of $\mu$. Similar results hold for the $L^q$ dimension and the entropy dimension of $\mu^{*n}$.