Skip to main content
SpringerLink
Log in

On Measures Resisting Multifractal Analysis

  • Chapter
  • First Online:
Nonlinear Dynamics New Directions

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 11))

Abstract

Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant measures invariant under linear maps of the circle.

Dedicated to Valentin Afraimovich on the occasion of his 65th birthday.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barreira, L., Pesin, Y., Schmeling, J.: On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. Chaos. 7, 27–38 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barreira, L., Saussol, B., Schmeling, J.: Higher-dimensional multifractal analysis. J. Math. Pures. Appl. 81, 67–91 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bowen, R.: Entropy for non-compact sets. Trans. Amer. Math. Soc. 184, 125–136 (1973)

    Article  MathSciNet  Google Scholar 

  4. Pesin, Y.: Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago (1997)

    Book  Google Scholar 

  5. Pesin, Y., Weiss, H.: The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples. Chaos. 7, 89–106 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Riedi, R.: Multifractal processes. In: Doukhan, P., et al. (eds.) Theory and Applications of Long-Range Dependence, pp. 625–716. Birkhäuser, Basel (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Centre for Mathematical Sciences, Lund Institute of Technology, Lund University, Box 118 SE-221 00, Lund, Sweden

    Jörg Schmeling

  2. LAMA, CNRS UMR 8050, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Stéphane Seuret

Authors
  1. Jörg Schmeling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stéphane Seuret
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jörg Schmeling .

Editor information

Editors and Affiliations

  1. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Hernán González-Aguilar

  2. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Edgardo Ugalde

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Schmeling, J., Seuret, S. (2015). On Measures Resisting Multifractal Analysis. In: González-Aguilar, H., Ugalde, E. (eds) Nonlinear Dynamics New Directions. Nonlinear Systems and Complexity, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-09867-8_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-09867-8_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-09866-1

  • Online ISBN: 978-3-319-09867-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation