Skip to main content
SpringerLink
Log in

Empirical multifractal moment measures of self-similar measure for q < 0*

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract.

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L q-spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L q- spectrum for q < 0 is generally considered significantly more difficult since the L q-spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L q-spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • L Olsen (2002) ArticleTitleEmpirical multifractal moment measures and moment scaling functions of self- similar multifractals Math Proc Camb Phil Soc 33 459–485 Occurrence Handle1919717

    MathSciNet  Google Scholar 

  • X Jiaqing W Min L Olsen (2008) ArticleTitleThe exact rate of convergence of the L q-spectrum of a self-similar measure for q < 0 J Math Anal 338 726–741 Occurrence Handle1132.28002 Occurrence Handle10.1016/j.jmaa.2007.05.042

    Article  MATH  Google Scholar 

  • L Olsen (1995) ArticleTitleA multifractal formalism Adv Math 116 82–195 Occurrence Handle0841.28012 Occurrence Handle10.1006/aima.1995.1066 Occurrence Handle1361481

    Article  MATH  MathSciNet  Google Scholar 

  • L Olsen (1998) ArticleTitleSelf-affine multifractal Sierpinski sponges in \({\Bbb R}^d\) Pacific J Math 183 143–199 Occurrence Handle0955.28004 Occurrence Handle1616626 Occurrence Handle10.2140/pjm.1998.183.143

    Article  MATH  MathSciNet  Google Scholar 

  • M Arbeiter N Patzschke (1996) ArticleTitleRandom self-similar multifractals Math Nachr 181 5–42 Occurrence Handle0873.28003 Occurrence Handle1409071

    MATH  MathSciNet  Google Scholar 

  • R Cawley RD Mauldin (1992) ArticleTitleMultifractal decomposition of Moran fractals Adv Math 92 196–236 Occurrence Handle0763.58018 Occurrence Handle10.1016/0001-8708(92)90064-R Occurrence Handle1155465

    Article  MATH  MathSciNet  Google Scholar 

  • W Feller (1986) An Introduction to Probability Theory and its Application EditionNumber3 NumberInSeries1 Wiley New York

    Google Scholar 

  • W Feller (1971) An Introduction to Probability Theory and its Application EditionNumber2 NumberInSeries2 Wiley New York

    Google Scholar 

  • S Graf (1995) ArticleTitleOn Bandt’s tangential distribution for self-similar measures Monatsh Math 120 223–246 Occurrence Handle0841.28011 Occurrence Handle10.1007/BF01294859 Occurrence Handle1363139

    Article  MATH  MathSciNet  Google Scholar 

  • J Hutchinson (1981) ArticleTitleFractals and self-similarity Indiana Univ Math J 30 713–747 Occurrence Handle0598.28011 Occurrence Handle10.1512/iumj.1981.30.30055 Occurrence Handle625600

    Article  MATH  MathSciNet  Google Scholar 

  • S Lalley (1988) ArticleTitleThe packing and covering function of some self-similar fractals Indiana Univ Math J 37 699–710 Occurrence Handle0665.28005 Occurrence Handle10.1512/iumj.1988.37.37034 Occurrence Handle962930

    Article  MATH  MathSciNet  Google Scholar 

  • Y Peres B Solomyak (2000) ArticleTitleExistence of L q dimensions and entropy dimension for self-conformal measures Indiana Univ Math J 49 1603–1621 Occurrence Handle0978.28004 Occurrence Handle10.1512/iumj.2000.49.1851 Occurrence Handle1838304

    Article  MATH  MathSciNet  Google Scholar 

  • D Rand (1989) ArticleTitleThe singularity spectrum f(α) for cookie-cutters Ergod Theor Dyn Syst 9 527–541 Occurrence Handle0664.58022 Occurrence Handle1016670 Occurrence Handle10.1017/S0143385700005162

    Article  MATH  MathSciNet  Google Scholar 

  • A Schief (1994) ArticleTitleSeparation properties for self-similar sets Proc Amer Math Soc 122 111–115 Occurrence Handle0807.28005 Occurrence Handle10.2307/2160849 Occurrence Handle1191872

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wuhan University of Technology, Wuhan, China

    Jiaqing Xiao

  2. South China University of Technology, Guangzhou, China

    Wu Min

Authors
  1. Jiaqing Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wu Min
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Authors’ addresses: Jiaqing Xiao, School of Science, Wuhan University of Technology, Wuhan 430070, China; Wu Min, School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiao, J., Min, W. Empirical multifractal moment measures of self-similar measure for q < 0* . Monatsh Math 156, 175–185 (2009). https://doi.org/10.1007/s00605-008-0563-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-008-0563-z

Search

Navigation