Skip to main content
SpringerLink
Log in

Dimension spectra of self-affine sets

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The dimension spectrumH(δ) is a function characterizing the distribution of dimension of sections. Using the multifractal formula for sofic measures, we show that the dimension spectra of irreducible self-affine sets (McMullen’s Carpet) coincide with the modified Legendre transform of the free energy Ψd(β). This variational relation leads to the formula of Hausdorff dimension of self-affine sets, max(δ +H(δ)) = Ψd(η), whereη is the logarithmic ratio of the contraction rates of the affine maps.

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

  1. P. Billingsley,Ergodic Theory and Information, Wiley, New York, 1965.

    MATH  Google Scholar 

  2. G. Brown, G. Michon and J. Peyrière,On the multifractal analysis of measures, Journal of Statistical Physics66 (1992), 775–790.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. A. Edgar and R. D. Mauldin,Multifractal decompositions of digraph recursive fractals, Proceedings of the London Mathematical Society65 (1992), 604–628.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Fujiwara, T. Hamachi and M. Oshikawa,Sofic systems and sofic measures, Surikaisekikenkyusho Kokyuroku552 (1985), 69–78 (in Japanese).

    Google Scholar 

  5. T. Kamae and S. Takahashi,Ergodic Theory and Fractals, Springer-Verlag, Tokyo, 1993 (in Japanese).

    Google Scholar 

  6. R. W. Kenyon and Y. Peres,Hausdorff dimensions of sofic affine-invariant sets, Israel Journal of Mathematics94 (1996), 157–178.

    Article  MathSciNet  Google Scholar 

  7. B. Kitchens and S. Tuncel,Finitary measures for subshifts of finite type and sofic systems, Memoirs of the American Mathematical Society58 (1985), 1–68.

    MathSciNet  Google Scholar 

  8. C. McMullen,The Hausdorff dimension of general Sierpiński carpets, Nagoya Mathematical Journal96 (1984), 1–9.

    MATH  MathSciNet  Google Scholar 

  9. S. Takahashi,Cellular automata and multifractals: dimension spectra of linear cellular automata, Physica D45 (1990), 36–48.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Takahashi,A variational formula for dimension spectra of linear cellular automata, Journal d’Analyse Mathématique64 (1994), 1–51.

    Article  MATH  Google Scholar 

  11. S. Takahashi,Multifractal formalism for sofic measures, preprint.

Download references

Author information

Authors and Affiliations

  1. Graduate School of Human Culture, Nara Women’s University, Kitauoya nishimachi, 630-8506, Nara, Japan

    Satoshi Takahashi

Authors
  1. Satoshi Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Takahashi, S. Dimension spectra of self-affine sets. Isr. J. Math. 127, 1–17 (2002). https://doi.org/10.1007/BF02784523

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02784523

Keywords

Search

Navigation