Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A simpler proof for the dimension of the graph of the classical Weierstrass function

Gerhard Keller

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Abstract

Let $W_{\lambda,b}(x)=\sum_{n=0}^{\infty}\lambda^{n}g(b^{n}x)$ where $b\geq2$ is an integer and $g(u)=\cos(2\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (In Symbolic Dynamics and Its Applications (1992) 285–293), Barański, Bárány and Romanowska (Adv. Math. 265 (2014) 32–59) and Tsujii (Nonlinearity 14 (2001) 1011–1027), we provide an elementary proof that the Hausdorff dimension of $W_{\lambda,b}$ equals $2+\frac{\log\lambda }{\log b}$ for all $\lambda\in(\lambda_{b},1)$ with a suitable $\lambda_{b}<1$. This reproduces results by Barański, Bárány and Romanowska (Adv. Math. 265 (2014) 32–59) without using the dimension theory for hyperbolic measures of Ledrappier and Young (Ann. of Math. (2) 122 (1985) 540–574; Comm. Math. Phys. 117 (1988) 529–548), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate.

Résumé

Soit $W_{\lambda,b}(x)=\sum_{n=0}^{\infty}\lambda^{n}g(b^{n}x)$, où $b\geq2$ est un nombre entier et $g(u)=\cos(2\pi u)$ (fonction de Weierstrass classique). En utilisant des idées et résultats de Ledrappier (In Symbolic Dynamics and Its Applications (1992) 285–293), de Barański, Bárány et Romanowska (Adv. Math. 265 (2014) 32–59) et de Tsujii (Nonlinearity 14 (2001) 1011–1027), nous présentons une démonstration élémentaire du fait que la dimension de Hausdorff de $W_{\lambda,b}$ est égale à $2+\frac{\log\lambda}{\log b}$ pour tout $\lambda\in(\lambda_{b},1)$ avec $\lambda_{b}<1$ approprié. Cela reproduit des résultats de Barański, Bárány et Romanowska (Adv. Math. 265 (2014) 32–59) sans utiliser la théorie de dimension des mesures hyperboliques de Ledrappier et Young (Ann. of Math. (2) 122 (1985) 540–574 ; Comm. Math. Phys. 117 (1988) 529–548), laquelle est remplacée par un argument téléscopique élémentaire conjointement avec une estimation récursive multi-échelle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 169-181.

Dates
Received: 16 April 2015
Revised: 16 July 2015
Accepted: 18 August 2015
First available in Project Euclid: 8 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1486544887

Digital Object Identifier
doi:10.1214/15-AIHP711

Mathematical Reviews number (MathSciNet)
MR3606737

Zentralblatt MATH identifier
06701696

Subjects
Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D45: Strange attractors, chaotic dynamics 37G35: Attractors and their bifurcations 37H20: Bifurcation theory [See also 37Gxx]

Keywords
Weierstrass function Hausdorff dimension

Citation

Keller, Gerhard. A simpler proof for the dimension of the graph of the classical Weierstrass function. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 169--181. doi:10.1214/15-AIHP711. https://projecteuclid.org/euclid.aihp/1486544887


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References

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