[Paper Review] Hausdorff dimension of the graphs of the classical Weierstrass functions

Weixiao Shen|arXiv (Cornell University)|May 15, 2015

Mathematical Dynamics and Fractals14 references10 citations

TL;DR

This paper proves that the Hausdorff dimension of the graph of the classical Weierstrass function $\sum_{n=0}^{\infty} \lambda^n \cos(2\pi b^n x)$ is exactly $2 + \log \lambda / \log b$ for integer $b \geq 2$ and $\lambda \in (1/b, 1)$, using a dynamical systems approach based on Ledrappier's criterion involving the local dimension of invariant measures. The result confirms a long-standing conjecture for this class of continuous, nowhere differentiable functions.

ABSTRACT

We show that the graph of the classical Weierstrass function $\sum_{n=0}^\infty λ^n \cos (2πb^n x)$ has Hausdorff dimension $2+\logλ/\log b$, for every integer $b\ge 2$ and every $λ\in (1/b,1)$. Replacing $\cos(2πx)$ by a general non-constant $C^2$ periodic function, we obtain the same result under a further assumption that $λb$ is close to $1$.

Motivation & Objective

• To resolve the long-standing conjecture that the Hausdorff dimension of the graph of the classical Weierstrass function equals $D = 2 + \log\lambda / \log b$.
• To extend this result beyond the cosine case to general $C^2$ $\mathbb{Z}$-periodic functions $\phi$ under a smallness condition on $\lambda b - 1$.
• To establish the dimension result via ergodic theory and the local dimension of invariant measures associated with an expanding endomorphism.
• To verify a modified transversality condition for the dynamical system framework, enabling the application of Ledrappier's criterion.

Proposed method

• Utilizes an expanding endomorphism $\Phi(x,y) = (bx \mod 1, (y - \hat{\phi}(x))/\lambda)$ on $\mathbb{R}/\mathbb{Z} \times \mathbb{R}$, whose invariant repeller is the graph of $f_{\lambda,b}^\phi$.
• Applies Ledrappier's theorem, which reduces the Hausdorff dimension problem to verifying that the local dimension of the measure $m_x$ is 1 for Lebesgue almost every $x \in (0,1)$.
• Defines the function $S(x,\mathbf{u}) = \sum_{n=1}^{\infty} \gamma^{n-1} \psi\left(\frac{x}{b^n} + \sum_{i=1}^n \frac{u_i}{b^i}\right)$ with $\gamma = 1/(\lambda b)$ and $\psi = \phi'$, representing the slope of strong unstable manifolds.
• Constructs the measure $m_x$ as the pushforward of the Bernoulli measure $\mathbb{P}$ on $\mathcal{A}^{\mathbb{Z}^+}$ via $\mathbf{u} \mapsto S(x,\mathbf{u})$.
• Employs a modified transversality condition and estimates on the decay of measure in dyadic cubes to control the local dimension of $m_x$.
• Uses Birkhoff's Ergodic Theorem and Egorov's Theorem to analyze typical orbits and derive lower bounds on the local dimension of the graph measure.

Experimental results

Research questions

• RQ1 Does the Hausdorff dimension of the Weierstrass function graph $W_{\lambda,b}$ equal $D = 2 + \log \lambda / \log b$ for integer $b \geq 2$ and $\lambda \in (1/b, 1)$?
• RQ2 Can the dimension result be extended to general $C^2$ $\mathbb{Z}$-periodic functions $\phi$ instead of just $\cos(2\pi x)$?
• RQ3 What conditions on $\lambda$ and $b$ ensure that the local dimension of the measure $m_x$ is 1 almost everywhere?
• RQ4 How can the transversality condition in Tsujii's framework be adapted to prove the dimension result for the full range of $\lambda \in (1/b, 1)$?

Key findings

• The Hausdorff dimension of the graph of $W_{\lambda,b}(x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi b^n x)$ is exactly $2 + \log \lambda / \log b$ for all integers $b \geq 2$ and $\lambda \in (1/b, 1)$.
• For any non-constant $C^2$ $\mathbb{Z}$-periodic function $\phi$, there exists $K_0 = K_0(\phi,b) > 1$ such that if $1 < \lambda b < K_0$, then the graph of $f_{\lambda,b}^\phi$ has Hausdorff dimension $D = 2 + \log \lambda / \log b$.
• The proof establishes that the local dimension of the measure $m_x$ is 1 for Lebesgue almost every $x \in (0,1)$, which by Ledrappier's theorem implies the dimension of the graph is $D$.
• The method confirms the conjecture for the classical Weierstrass function, which had remained open despite the box and packing dimensions being known to equal $D$.
• The result is obtained by verifying a modified transversality condition and using ergodic theory to control the decay of measure in dyadic cubes.