Inventiones mathematicae

, Volume 212, Issue 1, pp 109–132 | Cite as

Multifractal analysis of the Brjuno function

  • Stéphane Jaffard
  • Bruno Martin


The Brjuno function B is a 1-periodic, nowhere locally bounded function, introduced by Yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1. We show that \(T^p_\alpha \) regularity, introduced by Calderón and Zygmund, is the only one which is relevant in order to unfold the pointwise regularity properties of B; we determine its \(T^p_\alpha \) regularity at every point and show that it is directly related to the irrationality exponent \(\tau (x)\): its p-exponent at x is exactly \(1/\tau (x)\). This new example of multifractal function puts into light a new link between dynamical systems and fractal geometry. Finally we also determine the Hölder exponent of a primitive of B.

Mathematics Subject Classification

11A55 11J70 11K50 26A15 26A30 28A80 37F50 



The authors thank Yves Meyer and the anonymous referee for many remarks on previous versions of this text.


  1. 1.
    Abry, P., Jaffard, S., Leonarduzzi, R., Melot, C., Wendt, H.: Multifractal analysis based on \(p\)-exponents and lacunarity exponents. In: Fractal Geometry and Stochastics 70, Progr. Probab. Birkhäuser/Springer, pp. 279–313 (2015)Google Scholar
  2. 2.
    Abry, P., Jaffard, S., Leonarduzzi, R., Melot, C., Wendt, H.: New exponents for pointwise singularity classification. In: Seuret, S., Barral, J. (eds.) Proceedings of Fractals and Related Fields III, to appear (2017)Google Scholar
  3. 3.
    Balazard, M., Martin, B.: Comportement local moyen de la fonction de Brjuno. Fund. Math. 218, 193–224 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barral, J., Berestycki, J., Bertoin, J., Fan, A.H., Haas, B., Jaffard, S., Miermont, G., Peyrière, J.: Quelques interactions entre analyse, probabilités et fractals. Panoramas et Synthèses 32, Société Mathématique de France, Paris (2010)Google Scholar
  5. 5.
    Barral, J., Durand, A., Jaffard, S., Seuret, S.: Local multifractal analysis. In: Carfi, D., Lapidus, M.L., Pearse, E.P.J., van Frankenhuijsen, M. (Eds.) Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics. Contemporary Mathematics, AMS, 601, pp. 31–64 (2013)Google Scholar
  6. 6.
    Billingsley, P.: Ergodic Theory and Information. Wiley, New York (1965)MATHGoogle Scholar
  7. 7.
    Brjuno, A.D.: Analytic form of differential equations. I. Trudy Moskov. Mat. Obšč. 25, 119–262 (1971)Google Scholar
  8. 8.
    Brjuno, A.D.: Analytic form of differential equations. II. Trudy Moskov. Mat. Obšč. 26, 199–239 (1972)Google Scholar
  9. 9.
    Buff, X., Chéritat, A.: The Brjuno function continuously estimates the size of quadratic Siegel disks. Ann. Math. 164, 265–312 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Calderón, A.-P., Zygmund, A.: Local properties of solutions of elliptic partial differential equations. Stud. Math. 20, 171–225 (1961)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chamizo, F.: Automorphic forms and differentiability properties. Trans. Am. Math. Soc. 356, 1909–1935 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chamizo, F., Ubis, A.: Multifractal behavior of polynomial Fourier series. Adv. Math. 250, 1–3 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chamizo, F., Petrykiewicz, I., Ruiz-Cabello, I.: The Hölder exponent of some Fourier series. J. Fourier Anal. Appl. 1–20 (2016)Google Scholar
  14. 14.
    Cheraghi, D., Chéritat, A.: A proof of the Marmi–Moussa–Yoccoz conjecture for rotation numbers of high type. Invent. Math. 202(2), 677–742 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fraysse, A.: Generic validity of the multifractal formalism. SIAM J. Math. Anal. 39(2), 593–607 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jaffard, S.: Exposants de Hölder en des points donnés et coefficients d’ondelettes. C. R. Acad. Sci. Paris Sér. I Math. 308(4), 79–81 (1989)MathSciNetMATHGoogle Scholar
  17. 17.
    Jaffard, S.: The spectrum of singularities of Riemann’s function. Rev. Mat. Iberoamericana 12(2), 441–460 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jaffard, S.: On Davenport expansions. In: Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1. In: Proceedings of Symposium in Pure Mathematics, vol. 72. American Mathematical Society, Providence, RI, pp. 273–303 (2004)Google Scholar
  19. 19.
    Jaffard, S.: Wavelet techniques for pointwise regularity. Ann. Fac. Sci. Toul. 15(1), 3–33 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Jaffard, S., Mélot, C.: Wavelet analysis of fractal boundaries. Commun. Math. Phys. 258(3), 513–565 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Luzzi, L., Marmi, S., Nakada, H., Natsui, R.: Generalized Brjuno functions associated to \(\alpha \)-continued fractions. J. Approx. Theory 162(1), 24–41 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Marmi, S., Moussa, P., Yoccoz, J.-C.: The Brjuno functions and their regularity properties. Commun. Math. Phys. 186(2), 265–293 (1997)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Marmi, S., Moussa, P., Yoccoz, J.-C.: Complex Brjuno functions. J. Am. Math. Soc. 14(4), 783–841 (2001)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Marmi, S., Moussa, P., Yoccoz, J.-C.: Some properties of real and complex Brjuno functions. In: Cartier, P., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in Number Theory, Physics and Geometry I: On Random Matrices, Zeta Functions and Dynamical Systems, pp. 603–628. Springer, Berlin (2006)CrossRefGoogle Scholar
  25. 25.
    Meyer, Y.: Wavelets, vibrations and scalings, vol. 9. CRM Monograph Series. American Mathematical Society, Providence, RI (1998)Google Scholar
  26. 26.
    Nakada, H.: Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math. 4(2), 399–426 (1981)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pesin, Y.: Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics, Contemporary Views and Applications. University of Chicago Press, Chicago, IL (1997)CrossRefGoogle Scholar
  28. 28.
    Petrykiewicz, I.: Hölder regularity of arithmetic Fourier series arising from modular forms. arXiv:1311.0655
  29. 29.
    Petrykiewicz, I.: Differentiability of arithmetic Fourier series arising from Eisenstein series. Ramanujan J. 42(3), 527–581 (2017)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rivoal, T., Roques, J.: Convergence and modular type properties of a twisted Riemann series. Unif. Distrib. Theory 8(1), 97–119 (2013)MathSciNetMATHGoogle Scholar
  31. 31.
    Seuret, S., Ubis, A.: Local \({L}^2\)-regularity of Riemann’s Fourier series. Ann. Inst. Fourier (to appear)Google Scholar
  32. 32.
    Siegel, C.L.: Iteration of analytic functions. Ann. Math. 43(2), 607–612 (1942)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Yoccoz, J.-C.: Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Astérisque vol. 231. Petits diviseurs en dimension 1, 3–88 (1995)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, UPECUniversité Paris EstCréteilFrance
  2. 2.EA 2797, Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniversité du Littoral Côte d’OpaleCalaisFrance

Personalised recommendations