Skip to main content


  • Julien Barral (a1) (a2) and Stéphane Seuret (a3)

Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let $\{I_{w}\}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the finite dyadic words. Fix $\unicode[STIX]{x1D702}\in (0,1)$ , and a sequence $(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\unicode[STIX]{x1D702})}$ . We consider the (very sparse) remaining values $\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$ . We study the geometric and statistical information associated with $\widetilde{\unicode[STIX]{x1D707}}$ , and the relation between $\widetilde{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$ . To do so, we construct a random capacity $\mathsf{M}_{\unicode[STIX]{x1D707}}$ from $\widetilde{\unicode[STIX]{x1D707}}$ . This new object fulfills the multifractal formalism, and its free energy is closely related to that of  $\unicode[STIX]{x1D707}$ . Moreover, the free energy of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ generically exhibits one first order and one second order phase transition, while that of  $\unicode[STIX]{x1D707}$ is analytic. The geometry of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct $\unicode[STIX]{x1D707}$ from $\widetilde{\unicode[STIX]{x1D707}}$ by using the almost multiplicativity of $\unicode[STIX]{x1D707}$ and concatenation of words is discussed as well.

Hide All
1. Attia, N. and Barral, J., Hausdorff and packing spectra, large deviations, and free energy for branching random walks in ℝ d , Comm. Math. Phys. 331 (2014), 139187.
2. Barral, J., Mandelbrot cascades and related topics, in Geometry and Analysis of Fractals (ed. Feng, D.-J. and Lau, K.-S.), Springer Proceedings in Mathematics and Statistics (Springer, Berlin, Heidelberg, 2014).
3. Barral, J., Ben Nasr, F. and Peyrière, J., Comparing multifractal formalisms: the neighboring condition, Asian J. Math. 7 (2003), 149166.
4. Barral, J. and Seuret, S., Combining multifractal additive and multiplicative chaos, Comm. Math. Phys. 257(2) (2005), 473497.
5. Barral, J. and Seuret, S., Heterogeneous ubiquitous systems in ℝ d and Hausdorff dimensions, Bull. Braz. Math. Soc. (N.S.) 38(3) (2007), 467515.
6. Barral, J. and Seuret, S., Ubiquity and large intersections properties under digit frequencies constraints, Math. Proc. Cambridge Philos. Soc. 145(3) (2008), 527548.
7. Brown, G., Michon, G. and Peyrière, J., On the multifractal analysis of measures, J. Stat. Phys. 66 (1992), 775790.
8. Bruin, H. and Leplaideur, R., Renormalization, thermodynamic formalism and quasi-crystals in subshifts, Comm. Math. Phys. 321 (2013), 209247.
9. Bruin, H. and Leplaideur, R., Renormalization, freezing phase transitions and Fibonacci quasicristals, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), 739763, 2015.
10. Collet, P. and Koukiou, F., Large deviations for multiplicative chaos, Comm. Math. Phys. 147 (1992), 329342.
11. Collet, P., Lebowitz, J. L. and Porzio, A., The dimension spectrum of some dynamical systems, J. Stat. Phys. 47 (1987), 609644.
12. Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications (Jones and Bartlett Publishers, Boston, 1993).
13. Derrida, B. and Spohn, H., Polymers on disordered trees, spin glasses and traveling waves, J. Stat. Phys. 51 (1988), 817840.
14. Dodson, M., Melián, M., Pestana, D. and Vélani, S., Patterson measure and Ubiquity, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 3760.
15. Durand, A., Ubiquitous systems and metric number theory, Adv. Math. 218(2) (2008), 368394.
16. Fan, A. H., Feng, D. J. and Wu, J., Recurrence, dimension and entropy, J. Lond. Math. Soc. (2) 64(1) (2001), 229244.
17. Feng, D. J., Lyapounov exponents for products of matrices and multifractal analysis. Part I: positive matrices, Israël J. Math. 138 (2003), 353376.
18. Feng, D. J. and Lau, K. S., Pressure function for products of non-negative matrices, Math. Res. Lett. 9 (2002), 363378.
19. Feng, D.-J. and Olivier, E., Multifractal analysis of the weak Gibbs measures and phase transition- Application to some Bernoulli convolutions, Ergod. Th. & Dynam. Sys. 23 (2003), 17511784.
20. Heurteaux, Y., Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. H. Poincaré Probab. Stat. 34 (1998), 309338.
21. Hofbauer, F., Examples for the non uniqueness of the equilibrium state, Trans. Amer. Math. Soc. (1977), 223241.
22. Holley, R. and Waymire, E. C., Multifractal dimensions and scaling exponents for strongly bounded random fractals, Ann. Appl. Probab. 2 (1992), 819845.
23. Iommi, G. and Todd, M., Transience in dynamical systems, Ergod. Th. & Dynam. Sys. 33 (2013), 14501476.
24. Jaffard, S., On lacunary wavelet series, Ann. Appl. Probab. 10(1) (2000), 313329.
25. Jaffard, S., Wavelet techniques in multifractal analysis, in Fractal Geometry and Applications, Proceedings of Symposia in Pure Mathematics, Volume 72, (Part 2) pp. 91152 (American Mathematical Society, 2004).
26. Lévy Véhel, J. and Vojak, R., Multifractal analysis of Choquet capacities, Adv. Appl. Math. 20 (1998), 143.
27. Molchan, G. M., Scaling exponents and multifractal dimensions for independent random cascades, Comm. Math. Phys. 179 (1996), 681702.
28. Olsen, L., A multifractal formalism, Adv. Math. 116 (1995), 92195.
29. Rand, D. A., The singularity spectrum f (𝛼) for cookie-cutters, Ergod. Th. & Dynam. Sys. 9 (1989), 527541.
30. Ruelle, D., Thermodynamic formalism, in The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia of Mathematics and its Applications, Vol. 5, (Addison-Wesley Publishing Co., Reading, MA, 1978).
31. Sarig, O., Phase transitions for countable topological Markov shifts, Comm. Math. Phys. 217 (2001), 555577.
32. Schmeling, J., On the completeness of multifractal spectra, Ergod. Th. & Dynam. Sys. 19 (1999), 15951616.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed