Lq-spectrum of self-similar measures with overlaps in the absence of second-order identities

Sze-Man Ngai Hunnan Normal University and Georgia Southern University Yuanyuan Xie Hunan Normal University

Publications of CMSA of Harvard mathscidoc:1702.38001

For a self-similar measure in d-dimensional Euclidean space with overlaps but satisfies the so-called bounded measure type condition introduced by Tang and the authors, we set up a framework for deriving a closed formula for the Lq-spectrum of the measure for nonnegative q. The 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 Lq-spectrum have only been obtained earlier for measures satisfying Strichartz second-order identities. We illustrate how to use our results to prove the differentiability of the Lq-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.
Fractal, Lq-spectrum, multifractal formalism, self-similar measures, bounded measure type condition
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@inproceedings{sze-manlq-spectrum,
  title={Lq-spectrum of self-similar measures with overlaps in the absence of second-order identities},
  author={Sze-Man Ngai, and Yuanyuan Xie},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170205210203748078182},
}
Sze-Man Ngai, and Yuanyuan Xie. Lq-spectrum of self-similar measures with overlaps in the absence of second-order identities. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170205210203748078182.
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