## On the dimension of the graph of the classical Weierstrass function

Type:
Site:
Date:
16/01/2014 - 11:00 - 12:00
Salle:
Salle des thèses
Orateur:
ROMANOWSKA Julia
Localisation:
Université de Varsovie
Localisation:
Pologne
Résumé:

In my talk I will examine dimension of the graph of the famous Weierstrass non- differentiable function
$$W_{\lambda,b}(x)=\sum_{n=0}^\infty \lambda^n\cos(2\pi b^n x)$$
for an integer $b \ge 2$ and $1/b<\lambda<1$. In our recent paper, together with Balázs Bárány and Krzysztof Barański, we prove that for every $b$ there exists (explicitly given) $\lambda_b\in(1/b, 1)$ such that the Hausdorff dimension of the graph of $W_{\lambda,b}$ is equal to $D = 2 +\frac{\log \lambda}{\log b}$ for every $\lambda\in (\lambda_b , 1)$. We also show that the dimension is equal to $D$ for almost every $\lambda$ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function
$$f(x)=\sum_{n=0}^\infty\lambda^n\phi(b^nx)$$
for an integer $b\ge 2$ and $1/b < \lambda < 1$ is equal to $D$ for a typical $\mathbb{Z}$-periodic $C^3$ function $\phi$.

In my talk I will talk about these results as well as I will introduce Ledrappier-Young theory and results of Tsujii, which were used in the proofs.