Lq-SPECTRUM of SELF-SIMILAR MEASURES with OVERLAPS in the ABSENCE of SECOND-ORDER IDENTITIES

Sze Man Ngai, Yuanyuan Xie

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

For the class of self-similar measures in Rd with overlaps that are essentially of finite type, we set up a framework for deriving a closed formula for the Lq-spectrum of the measure for q ≥ 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 Lq-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 Lq-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.

Original languageEnglish
Pages (from-to)56-103
Number of pages48
JournalJournal of the Australian Mathematical Society
Volume106
Issue number1
DOIs
StatePublished - Feb 1 2019

Scopus Subject Areas

  • General Mathematics

Keywords

  • essentially of finite type
  • fractal
  • L -spectrum
  • multifractal formalism
  • self-similar measure

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