Geodesic
The thermodynamic formalism for the de~Rham function: increment method
Izvestiya. Mathematics , Tome 76 (2012) no. 3, pp. 431-445.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the de Rham function: the unique continuous (nowhere differentiable) function $F \in L^1(\mathbb{R})$ with $\int F(x)\,dx=1$ satisfying the functional equation $F(x)=F(3x)+\frac{1}{3}\bigl(F(3x-1)+F(3x+1) \bigr)+\frac{2}{3}\bigl(F(3x-2)+F(3x+2)\bigr)$. We show that its pointwise Hölder regularity $\alpha(x)=\liminf_{h\to 0}\frac{\log(|F(x+h)-F(x)|)}{\log |h|}$ differs widely from point to point, and the values of $\alpha(x)$ fill an interval parametrizing the fractal sets $E^{(\alpha)}$, where $E^{(\alpha)}$ is the set of points $x$ with Hölder exponent $\alpha(x)=\alpha$. We also prove that the thermodynamic formalism (increment method) holds for the de Rham function: we have a heuristic formula $d(\alpha)=\inf_{q >0}(\alpha q-\zeta(q)+1)$ relating the order of decay of $\int_{\mathbb{R}}|F(x+h)-F(x)|^{q}\,dx \sim |h|^{\zeta(q)}$ as $h \to 0$ with the Hausdorff dimension $d(\alpha)$ of $E^{(\alpha)}$.
Keywords: Hölder regularity, thermodynamic formalism.
Mots-clés : Hausdorff dimension, increments 
@article{IM2_2012_76_3_a0,
     author = {M. Ben Slimane},
     title = {The thermodynamic formalism for the {de~Rham} function: increment method},
     journal = {Izvestiya. Mathematics },
     pages = {431--445},
     publisher = {mathdoc},
     volume = {76},
     number = {3},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a0/}
}
TY  - JOUR
AU  - M. Ben Slimane
TI  - The thermodynamic formalism for the de~Rham function: increment method
JO  - Izvestiya. Mathematics 
PY  - 2012
SP  - 431
EP  - 445
VL  - 76
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a0/
LA  - en
ID  - IM2_2012_76_3_a0
ER  - 
%0 Journal Article
%A M. Ben Slimane
%T The thermodynamic formalism for the de~Rham function: increment method
%J Izvestiya. Mathematics 
%D 2012
%P 431-445
%V 76
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a0/
%G en
%F IM2_2012_76_3_a0
M. Ben Slimane. The thermodynamic formalism for the de~Rham function: increment method. Izvestiya. Mathematics , Tome 76 (2012) no. 3, pp. 431-445. http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a0/

[1] R. Benzi, G. Paladin, G. Parisi, A. Vulpiani, “On the multifractal nature of fully developed turbulence and chaotic systems”, J. Phys. A, 17:18 (1984), 3521–3531 | DOI | MR

[2] B. B. Mandelbrot, “Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier”, J. Fluid Mech., 62:2 (1974), 331–358 | DOI | Zbl

[3] J.-P.. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors”, Rev. Modern Phys., 57:3 (1985), 617–656 | DOI | MR | Zbl

[4] T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets”, Phys. Rev. A (3), 33:2 (1986), 1141–1151 | DOI | MR | Zbl

[5] U. Frisch, G. Parisi, “Fully developped turbulence and intermittency”, Proc. Int. Summer school Phys. Enrico Fermi, North Holland, Amsterdam, 1985, 84–88

[6] E. Bacry, J. F. Muzy, A. Arnéodo, “Singularity spectrum of fractal signals from wavelet analysis: exact results”, J. Statist. Phys., 70:3–4 (1993), 635–674 | DOI | MR | Zbl

[7] J. Aouidi, M. Ben Slimane, “Multi-fractal formalism for quasi-self-similar functions”, J. Statist. Phys., 108:3–4 (2002), 541–590 | DOI | MR | Zbl

[8] S. Jaffard, “Multifractal formalism for functions. Part 1: Results valid for all functions; Part 2: Self-similar functions”, SIAM J. Math. Anal., 28:4 (1997), 944–998 | DOI | DOI | MR | MR | Zbl

[9] S. Jaffard, “On the Frisch–Parisi conjecture”, J. Math. Pures Appl. (9), 79:6 (2000), 525–552 | DOI | MR | Zbl

[10] A. Fraysse, “Generic validity of the multifractal formalism”, SIAM J. Math. Anal., 37:2 (2007), 593–607 | DOI | MR | Zbl

[11] I. Daubechies, J. C. Lagarias, “On the thermodynamic formalism for multifractal functions”, Rev. Math. Phys., 6:5A (1994), 1033–1070 | DOI | MR | Zbl

[12] G. De Rham, “Sur une courbe plane”, J. Math. Pures Appl. (9), 35 (1956), 25–42 | MR | Zbl

[13] G. De Rham, “Sur un exemple de fonction continue sans dérivée”, Enseignement Math. (2), 3 (1957), 71–72 | MR | Zbl

[14] I. Daubechies, J. C. Lagarias, “Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals”, SIAM J. Math. Anal., 22:4 (1992), 1031–1079 | DOI | MR | Zbl

[15] Y. Meyer, “Fonctions multifractales”, Ecole Normale Supérieure de Cachan, France

[16] S. Seuret, “On multifractality and time subordination for continuous functions”, Adv. Math., 220:3 (2009), 936–963 | DOI | MR | Zbl

[17] M. Ben Slimane, “Multifractal formalism for the generalized de Rham function”, Curr. Dev. Theory Appl. Wavelets, 2:1 (2008), 45–88 | MR | Zbl

[18] V. Yu. Protasov, “On the regularity of de Rham curves”, Izv. Math., 68:3 (2004), 567–606 | DOI | MR

[19] I. Daubechies, J. C. Lagarias, “Two-scale difference equations. I. Existence and global regularity of solutions”, SIAM J. Math. Anal., 22:5 (1991), 1388–1410 | DOI | MR | Zbl

[20] H. G. Eggleston, “The fractional dimension of a set defined by decimal properties”, Quart. J. Math., Oxford Ser., 20 (1949), 31–36 | DOI | MR | Zbl

[21] J. E. Hutchinson, “Fractals and self-similarity”, Indiana Univ. Math. J., 30:5 (1981), 713–747 | DOI | MR | Zbl

[22] K. Falconer, Fractal geometry: mathematical foundations and applications, Wiley, New York, 1990 | MR | Zbl