## Real Analysis Exchange

- Real Anal. Exchange
- Volume 38, Number 2 (2012), 391-408.

### Multi-Fractal Analysis of Convolution Powers of Measures

Cameron Bruggeman and Kathryn E. Hare

#### 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 \(q\geq 1\).

#### Article information

**Source**

Real Anal. Exchange, Volume 38, Number 2 (2012), 391-408.

**Dates**

First available in Project Euclid: 27 June 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1403894899

**Mathematical Reviews number (MathSciNet)**

MR3261884

**Zentralblatt MATH identifier**

1298.28013

**Subjects**

Primary: 28A80: Fractals [See also 37Fxx]

Secondary: 42A85: Convolution, factorization

**Keywords**

multi-fractal analysis local dimension \(L^q\)-spectrum self-similar measure

#### Citation

Bruggeman, Cameron; Hare, Kathryn E. Multi-Fractal Analysis of Convolution Powers of Measures. Real Anal. Exchange 38 (2012), no. 2, 391--408. https://projecteuclid.org/euclid.rae/1403894899