Convolutions of equicontractive self-similar measures on the line
Let \(\mu\) be an equicontractive self-similar measure on the real line, that is, \(\mu\) satisfies \(\mu=\sum_{j=1}^{m}p_j\mu\circ\phi_j^{-1}\) for some positive probability vector \((p_j)_{j=1,\ldots,m}\), where \(\{\phi_j\}_{j=1,\ldots,m}\) is a system of contractive similitudes on \({\mathbb R}\), all of them with a common contraction ratio \(0 <\rho <1\). The authors prove that the Hausdorff dimension of the \(n\)-fold convolution measure \(\mu^{*n}\) increasingly tends to unity when \(n\) tends to infinity. They further establish that both the entropy dimension and the \(L^q\)-dimensions (for \(1< q\leq 2\)) of \(\mu^{*n}\) also have that asymptotic behavior.
scientific article; zbMATH DE number 1952922