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$L^{q}$-SPECTRUM OF SELF-SIMILAR MEASURES WITH OVERLAPS IN THE ABSENCE OF SECOND-ORDER IDENTITIES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 106 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 22 August 2018, pp. 56-103
- Print publication:
- February 2019
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- Article
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For the class of self-similar measures in
$\mathbb{R}^{d}$ with overlaps that are essentially of finite type, we set up a framework for deriving a closed formula for the 
$L^{q}$-spectrum of the measure for 
$q\geq 0$. This framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the 
$L^{q}$-spectrum have only been obtained earlier for measures satisfying Strichartz’s second-order identities. We illustrate how to use our results to prove the differentiability of the 
$L^{q}$-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.
