Multi-Fractal Analysis of Convolution Powers of Measures

Abstract

We investigate the multi-fractal analysis of (large) convolution powers of probability measures on \(\mathbb{R}\). If the measure \(\mu \) satisfies \((N)\) supp\(\mu =[0,N]\) for some \(N\), then under weak assumptions there is an isolated point in the multi-fractal spectrum of \(\mu ^{n}\) for sufficiently large \(n\). A formula is found for the limiting behaviour (as \(n\rightarrow \infty \)) of the \(L^{q}\)-spectrum of \(\mu ^{n}\) and this is related to the limit of the energy dimension of \(\mu ^{n}\) when \(q\geq 1\).