Skip to main content
SpringerLink
Log in

Spectral Gap Property for Random Dynamics on the Real Line and Multifractal Analysis of Generalised Takagi Functions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the random iteration of finitely many expanding \(\mathscr {C}^{1+\epsilon }\) diffeomorphisms on the real line without a common fixed point. We derive the spectral gap property of the associated transition operator acting on spaces of Hölder continuous functions. As an application we introduce generalised Takagi functions on the real line and we perform a complete multifractal analysis of the pointwise Hölder exponents of these functions.

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. Allaart, P.C., Kawamura, K.: Extreme values of some continuous nowhere differentiable functions. Math. Proc. Cambridge Philos. Soc. 140(2), 269–295 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  2. Allaart, P.C.: Differentiability and Hölder spectra of a class of self-affine functions. Adv. Math. 328, 1–39 (2018)

    Article  MathSciNet  Google Scholar 

  3. Barany, B., Kiss, G., Kolossvary, I.: Pointwise regularity of parameterized affine zipper fractal curves. Nonlinearity 31, 1705–1733 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  4. Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications, 2nd edn. Wiley, Hoboken (2003)

    Book  Google Scholar 

  5. Hata, M., Yamaguti, M.: The Takagi function and its generalization. Jpn. J. Appl. Math. 1(1), 183–199 (1984)

    Article  MathSciNet  Google Scholar 

  6. Ionescu Tulcea, C .T., Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètement continues. Ann. Math. (2) 52, 140–147 (1950)

    Article  MathSciNet  Google Scholar 

  7. Jordan, T., Kesseböhmer, M., Pollicott, M., Stratmann, B.O.: Sets of nondifferentiability for conjugacies between expanding interval maps. Fund. Math. 206, 161–183 (2009)

    Article  MathSciNet  Google Scholar 

  8. Jaerisch, J., Sumi, H.: Multifractal formalism for expanding rational semigroups and random complex dynamical systems. Nonlinearity 28, 2913–2938 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  9. Jaerisch, J., Sumi, H.: Pointwise Hölder exponents of the complex analogues of the Takagi function in random complex dynamics. Adv. Math. 313, 839–874 (2017)

    Article  MathSciNet  Google Scholar 

  10. Jaerisch, J., Sumi, H.: Multifractal analysis of generalised Takagi functions on the real line, to appear in RIMS Kokyuroku

  11. Kato, T.: Perturbation theory for linear operators, second ed., Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, Band 132

  12. Kesseböhmer, M., Stratmann, B.O.: Fractal analysis for sets of non-differentiability of Minkowski’s question mark function. J. Number Theory 128(9), 2663–2686 (2008)

    Article  MathSciNet  Google Scholar 

  13. Kesseböhmer, M., Stratmann, B.O.: Hölder-differentiability of Gibbs distribution functions. Math. Proc. Camb. Philos. Soc. 147(2), 489–503 (2009)

    Article  Google Scholar 

  14. Ljubich, M.J.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dynam. Systems 3(3), 351–385 (1983)

    Article  MathSciNet  Google Scholar 

  15. Mauldin, R.D., Urbański, M.: Graph Directed Markov Systems. Cambridge Tracts in Mathematics, vol. 148. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  16. Mauldin, R.D., Williams, S.C.: On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298(2), 793–803 (1986)

    Article  MathSciNet  Google Scholar 

  17. Patzschke, N.: Self-conformal multifractal measures. Adv. Appl. Math. 19(4), 486–513 (1997)

    Article  MathSciNet  Google Scholar 

  18. Pesin, Y.B.: Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics. Contemporary Views and Applications. University of Chicago Press, Chicago (1997)

    Book  Google Scholar 

  19. Rockafellar, R.T.: Convex Analysis, Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)

    Google Scholar 

  20. Schmeling, J.: On the completeness of multifractal spectra. Ergodic Theory Dynam. Syst. 19(6), 1595–1616 (1999)

    Article  MathSciNet  Google Scholar 

  21. Sekiguchi, T., Shiota, Y.: A generalization of Hata–Yamaguti’s results on the Takagi function. Jpn. J. Indust. Appl. Math. 8(2), 203–219 (1991)

    Article  MathSciNet  Google Scholar 

  22. Sumi, H.: Random complex dynamics and semigroups of holomorphic maps. Proc. London Math. Soc. 1(102), 50–112 (2011)

    Article  MathSciNet  Google Scholar 

  23. Sumi, H.: Cooperation principle, stability and bifurcation in random complex dynamics. Adv. Math. 245, 137–181 (2013)

    Article  MathSciNet  Google Scholar 

  24. Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Rich Stankewitz for valuable comments. The research of the first author was partially supported by JSPS Kakenhi 15H06416, 17K14203. The research of the second author was partially supported by JSPS Kakenhi 15K04899, 18H03671.

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

    Johannes Jaerisch

  2. Course of Mathematical Science, Department of Human Coexistence Graduate School of Human and Environmental Studies, Kyoto University Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto, 606-8501, Japan

    Hiroki Sumi

Authors
  1. Johannes Jaerisch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroki Sumi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johannes Jaerisch.

Additional information

Communicated by C. Liverani

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jaerisch, J., Sumi, H. Spectral Gap Property for Random Dynamics on the Real Line and Multifractal Analysis of Generalised Takagi Functions . Commun. Math. Phys. 377, 1–36 (2020). https://doi.org/10.1007/s00220-020-03766-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-020-03766-5

Search

Navigation