Hausdorff dimension of the graphs of the classical Weierstrass functions

In this paper, we obtain the explicit value of the Hausdorff dimension of the graphs of the classical Weierstrass functions, by proving absolute continuity of the SRB measures of the associated solenoidal attractors.

1. Introduction

In Real Analysis, the classical Weierstrass function is

$\displaystyle W_{\lambda,b}(x) = \sum\limits_{n=0}^{\infty} \lambda^n \cos(2\pi b^n x)$

with ${1/b < \lambda < 1}$ .

Note that the Weierstrass functions have the form

$\displaystyle f^{\phi}_{\lambda,b}(x) = \sum\limits_{n=0}^{\infty} \lambda^n \phi(b^n x)$

where ${\phi}$ is a ${\mathbb{Z}}$ -periodic ${C^2}$ -function.

Weierstrass (1872) and Hardy (1916) were interested in ${W_{\lambda,b}}$ because they are concrete examples of continuous but nowhere differentiable functions.

Remark 1 The graph of ${f^{\phi}_{\lambda,b}}$ tends to be a “fractal object” because ${f^{\phi}_{\lambda,b}}$ is self-similar in the sense that

$\displaystyle f^{\phi}_{\lambda, b}(x) = \phi(x) + \lambda f^{\phi}_{\lambda,b}(bx)$

We will come back to this point later.

Remark 2 ${f^{\phi}_{\lambda,b}}$ is a ${C^{\alpha}}$ -function for all ${0\leq \alpha < \frac{-\log\lambda}{\log b}}$ . In fact, for all ${x,y\in[0,1]}$ , we have

$\displaystyle \frac{f^{\phi}_{\lambda, b}(x) - f^{\phi}_{\lambda,b}(y)}{|x-y|^{\alpha}} = \sum\limits_{n=0}^{\infty} \lambda^n b^{n\alpha} \left(\frac{\phi(b^n x) - \phi(b^n y)}{|b^n x - b^n y|^{\alpha}}\right),$

so that

$\displaystyle \frac{f^{\phi}_{\lambda, b}(x) - f^{\phi}_{\lambda,b}(y)}{|x-y|^{\alpha}} \leq \|\phi\|_{C^{\alpha}} \sum\limits_{n=0}^{\infty}(\lambda b^{\alpha})^n:=C(\phi,\alpha,\lambda,b) < \infty$

whenever ${\lambda b^{\alpha} < 1}$ , i.e., ${\alpha < -\log\lambda/\log b}$ .

The study of the graphs of ${W_{\lambda,b}}$ as fractal sets started with the work of Besicovitch-Ursell in 1937.

Remark 3 The Hausdorff dimension of the graph of a ${C^{\alpha}}$ -function ${f:[0,1]\rightarrow\mathbb{R}}$ is

$\displaystyle \textrm{dim}(\textrm{graph}(f))\leq 2 - \alpha$

Indeed, for each ${n\in\mathbb{N}}$ , the Hölder continuity condition

$\displaystyle |f(x)-f(y)|\leq C|x-y|^{\alpha}$

leads us to the “natural cover” of ${G=\textrm{graph}(f)}$ by the family ${(R_{j,n})_{j=1}^n}$ of rectangles given by

$\displaystyle R_{j,n}:=\left[\frac{j-1}{n}, \frac{j}{n}\right] \times \left[f(j/n)-\frac{C}{n^{\alpha}}, f(j/n)+\frac{C}{n^{\alpha}}\right]$

Nevertheless, a direct calculation with the family ${(R_{j,n})_{j=1}^n}$ does not give us an appropriate bound on ${\textrm{dim}(G)}$ . In fact, since ${\textrm{diam}(R_{j,n})\leq 4C/n^{\alpha}}$ for each ${j=1,\dots, n}$ , we have

$\displaystyle \sum\limits_{j=1}^n\textrm{diam}(R_{j,n})^d\leq n\left(\frac{4C}{n^{\alpha}}\right)^d = (4C)^{1/\alpha} < \infty$

for ${d=1/\alpha}$ . Because ${n\in\mathbb{N}}$ is arbitrary, we deduce that ${\textrm{dim}(G)\leq 1/\alpha}$ . Of course, this bound is certainly suboptimal for ${\alpha<1/2}$ (because we know that ${\textrm{dim}(G)\leq 2 < 1/\alpha}$ anyway).Fortunately, we can refine the covering ${(R_{j,n})}$ by taking into account that each rectangle ${R_{j,n}}$ tends to be more vertical than horizontal (i.e., its height ${2C/n^{\alpha}}$ is usually larger than its width ${1/n}$ ). More precisely, we can divide each rectangle ${R_{j,n}}$ into ${\lfloor n^{1-\alpha}\rfloor}$ squares, say

$\displaystyle R_{j,n} = \bigcup\limits_{k=1}^{\lfloor n^{1-\alpha}\rfloor}Q_{j,n,k},$

such that every square ${Q_{j,n,k}}$ has diameter ${\leq 2C/n}$ . In this way, we obtain a covering ${(Q_{j,n,k})}$ of ${G}$ such that

$\displaystyle \sum\limits_{j=1}^n\sum\limits_{k=1}^{\lfloor n^{1-\alpha}\rfloor} \textrm{diam}(Q_{j,n,k})^d \leq n\cdot n^{1-\alpha}\cdot\left(\frac{2}{n}\right)^d\leq (2C)^{2-\alpha}<\infty$

for ${d=2-\alpha}$ . Since ${n\in\mathbb{N}}$ is arbitrary, we conclude the desired bound

$\displaystyle \textrm{dim}(G)\leq 2-\alpha$

A long-standing conjecture about the fractal geometry of ${W_{\lambda,b}}$ is:

Conjecture (Mandelbrot 1977): The Hausdorff dimension of the graph of ${W_{\lambda,b}}$ is

$\displaystyle 1<\textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b} < 2$

Remark 4 In view of remarks 2 and 3, the whole point of Mandelbrot’s conjecture is to establish the lower bound

$\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) \geq 2 + \frac{\log\lambda}{\log b}$

Remark 5 The analog of Mandelbrot conjecture for the box and packing dimensions is known to be true: see, e.g., these papers here and here).

In a recent paper (see here), Shen proved the following result:

Theorem 1 (Shen) For any ${b\geq 2}$ integer and for all ${1/b < \lambda < 1}$ , the Mandelbrot conjecture is true, i.e.,

$\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}$

Remark 6 The techniques employed by Shen also allow him to show that given ${\phi:\mathbb{R}\rightarrow\mathbb{R}}$ a ${\mathbb{Z}}$ -periodic, non-constant, ${C^2}$ function, and given ${b\geq 2}$ integer, there exists ${K=K(\phi,b)>1}$ such that

$\displaystyle \textrm{dim}(\textrm{graph}(f^{\phi}_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}$

for all ${1/K < \lambda < 1}$ .

Remark 7 A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all ${b\geq 2}$ integer, there exists ${1/b < \lambda_b < 1}$ such that

$\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}$

for all ${\lambda_b < \lambda < 1}$ .

The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem1 by discussing the particular case when ${1/b<\lambda<2/b}$ and ${b\in\mathbb{N}}$ is large.

2. Ledrappier’s dynamical approach

If ${b\geq 2}$ is an integer, then the self-similar function ${f^{\phi}_{\lambda,b}}$ (cf. Remark 1) is also ${\mathbb{Z}}$ -periodic, i.e., ${f^{\phi}_{\lambda,b}(x+1) = f^{\phi}_{\lambda,b}(x)}$ for all ${x\in\mathbb{R}}$ . In particular, if ${b\geq 2}$ is an integer, then ${\textrm{graph}(f^{\phi}_{\lambda,b})}$ is an invariant repeller for the endomorphism ${\Phi:\mathbb{R}/\mathbb{Z}\times\mathbb{R}\rightarrow \mathbb{R}/\mathbb{Z}\times\mathbb{R}}$ given by

$\displaystyle \Phi(x,y) = \left(bx\textrm{ mod }1, \frac{y-\phi(x)}{\lambda}\right)$

This dynamical characterization of ${G = \textrm{graph}(f^{\phi}_{\lambda,b})}$ led Ledrappier to the following criterion for the validity of Mandelbrot’s conjecture when ${b\geq 2}$ is an integer.

Denote by ${\mathcal{A}}$ the alphabet ${\mathcal{A}=\{0,\dots,b-1\}}$ . The unstable manifolds of ${\Phi}$ through ${G}$ have slopes of the form

$\displaystyle (1,-\gamma \cdot s(x,u))$

where ${\frac{1}{b} < \gamma = \frac{1}{\lambda b} <1}$ , ${x\in\mathbb{R}}$ , ${u\in\mathcal{A}^{\mathbb{N}}}$ , and

$\displaystyle s(x,u):=\sum\limits_{n=0}^{\infty} \gamma^n \phi'\left(\frac{x + u_1 + u_2 b + \dots + u_n b^{n-1}}{b^n}\right)$

In this context, the push-forwards ${m_x := (u\mapsto s(x,u))_*\mathbb{P}}$ of the Bernoulli measure ${\mathbb{P}}$ on ${\mathcal{A}^{\mathbb{N}}}$ (induced by the discrete measure assigning weight ${1/b}$ to each letter of the alphabet ${\mathcal{A}}$ ) play the role of conditional measures along vertical fibers of the unique Sinai-Ruelle-Bowen (SRB) measure ${\theta}$ of the expanding endomorphism ${T:\mathbb{R}/\mathbb{Z}\times\mathbb{R} \rightarrow \mathbb{R}/\mathbb{Z}\times\mathbb{R}}$ ,

$\displaystyle T(x,y) = (bx\textrm{ mod }1, \gamma y + \psi(x)),$

where ${\gamma=1/\lambda b}$ and ${\psi(x)=\phi'(x)}$ . In plain terms, this means that

$\displaystyle \theta = \int_{\mathbb{R}/\mathbb{Z}} m_x \, d\textrm{Leb}(x) \ \ \ \ \ (1)$

where ${\theta}$ is the unique ${T}$ -invariant probability measure which is absolutely continuous along unstable manifolds (see Tsujii’s paper).

As it was shown by Ledrappier in 1992, the fractal geometry of the conditional measures ${m_x}$ have important consequences for the fractal geometry of the graph ${G}$ :

Theorem 2 (Ledrappier) Suppose that for Lebesgue almost every ${x\in\mathbb{R}}$ the conditional measures ${m_x}$ have dimension ${\textrm{dim}(m_x)=1}$ , i.e.,

$\displaystyle \lim\limits_{r\rightarrow 0}\frac{\log m_x(B(z,r))}{\log r} = 1 \textrm{ for } m_x\textrm{-a.e. } z$

Then, the graph ${G=\textrm{graph}(f^{\phi}_{\lambda,b})}$ has Hausdorff dimension

$\displaystyle \textrm{dim}(G) = 2 + \frac{\log\lambda}{\log b}$

Remark 8 Very roughly speaking, the proof of Ledrappier theorem goes as follows. By Remark 4, it suffices to prove that ${\textrm{dim}(G)\geq 2 + \frac{\log\lambda}{\log b}}$ . By Frostman lemma, we need to construct a Borel measure ${\nu}$ supported on ${G}$ such that

$\displaystyle \underline{\textrm{dim}}(\nu) := \textrm{ ess }\inf \underline{d}(\nu,x) \geq 2 + \frac{\log\lambda}{\log b}$

where ${\underline{d}(\nu,x):=\liminf\limits_{r\rightarrow 0}\log \nu(B(x,r))/\log r}$ . Finally, the main point is that the assumptions in Ledrappier theorem allow to prove that the measure ${\mu^{\phi}_{\lambda, b}}$ given by the lift to ${G}$ of the Lebesgue measure on ${[0,1]}$ via the map ${x\mapsto (x,f^{\phi}_{\lambda,b}(x))}$ satisfies

$\displaystyle \underline{\textrm{dim}}(\mu^{\phi}_{\lambda,b}) \geq 2 + \frac{\log\lambda}{\log b}$

An interesting consequence of Ledrappier theorem and the equation 1 is the following criterion for Mandelbrot’s conjecture:

Corollary 3 If ${\theta}$ is absolutely continuous with respect to the Lebesgue measure ${\textrm{Leb}_{\mathbb{R}^2}}$ , then

$\displaystyle \textrm{dim}(G) = 2 + \frac{\log\lambda}{\log b}$

Proof: By (1), the absolute continuity of ${\theta}$ implies that ${m_x}$ is absolutely continuous with respect to ${\textrm{Leb}_{\mathbb{R}}}$ for Lebesgue almost every ${x\in\mathbb{R}}$ .

Since ${m_x\ll \textrm{Leb}_{\mathbb{R}}}$ for almost every ${x}$ implies that ${\textrm{dim}(m_x)=1}$ for almost every ${x}$ , the desired corollary now follows from Ledrappier’s theorem. $\Box$

3. Tsujii’s theorem

The relevance of Corollary 3 is explained by the fact that Tsujii found an explicittransversality condition implying the absolute continuity of ${\theta}$ .

More precisely, Tsujii firstly introduced the following definition:

Definition 4

Given ${\varepsilon>0}$ , ${\delta>0}$ and ${x_0\in\mathbb{R}/\mathbb{Z}}$ , we say that two infinite words ${u, v\in\mathcal{A}^{\mathbb{N}}}$ are ${(\varepsilon,\delta)}$ -transverse at ${x_0}$ if either
$\displaystyle |s(x_0,u)-s(x_0,v)|>\varepsilon$

or

$\displaystyle |s'(x_0,u)-s'(x_0,v)|>\delta$

Given ${q\in\mathbb{N}}$ , ${\varepsilon>0}$ , ${\delta>0}$ and ${x_0\in\mathbb{R}/\mathbb{Z}}$ , we say that two finite words ${k,l\in\mathcal{A}^q}$ are ${(\varepsilon,\delta)}$ -transverse at ${x_0}$ if ${ku}$ , ${lv}$ are ${(\varepsilon,\delta)}$ -transverse at ${x_0}$ for all pairs of infinite words ${u,v\in\mathcal{A}^{\mathbb{N}}}$ ; otherwise, we say that ${k}$ and ${l}$ are ${(\varepsilon,\delta)}$ -tangent at ${x_0}$ ;

${E(q,x_0;\varepsilon,\delta):= \{(k,l)\in\mathcal{A}^q\times\mathcal{A}^q: (k,l) \textrm{ is } (\varepsilon,\delta)\textrm{-tangent at } x_0\}}$

${E(q,x_0):=\bigcap\limits_{\varepsilon>0}\bigcap\limits_{\delta>0} E(q,x_0;\varepsilon,\delta)}$ ;

${e(q,x_0):=\max\limits_{k\in\mathcal{A}^q}\#\{l\in\mathcal{A}^q: (k,l)\in E(q,x_0)\}}$

${e(q):=\max\limits_{x_0\in\mathbb{R}/\mathbb{Z}} e(q,x_0)}$ .

Next, Tsujii proves the following result:

Theorem 5 (Tsujii) If there exists ${q\geq 1}$ integer such that ${e(q)<(\gamma b)^q}$ , then

$\displaystyle \theta\ll\textrm{Leb}_{\mathbb{R}^2}$

Remark 9 Intuitively, Tsujii’s theorem says the following. The transversality condition ${e(q)<(\gamma b)^q}$ implies that the majority of strong unstable manifolds ${\ell^{uu}}$ are mutually transverse, so that they almost fill a small neighborhood ${U}$ of some point ${x_0}$ (see the figure below extracted from this paper of Tsujii). Since the SRB measure ${\theta}$ is absolutely continuous along strong unstable manifolds, the fact that the ${\ell^{uu}}$ ‘s almost fill ${U}$ implies that ${\theta}$ becomes “comparable” to the restriction of the Lebesgue measure ${\textrm{Leb}_{\mathbb{R}^2}}$ to ${U}$ .

Remark 10 In this setting, Barańsky-Barány-Romanowska obtained their main result by showing that, for adequate choices of the parameters ${\lambda}$ and ${b}$ , one has ${e(1)=1}$ . Indeed, once we know that ${e(1)=1}$ , since ${1<\gamma b}$ , they can apply Tsujii’s theorem and Ledrappier’s theorem (or rather Corollary 3) to derive the validity of Mandelbrot’s conjecture for certain parameters ${\lambda}$ and ${b}$ .

For the sake of exposition, we will give just a flavor of the proof of Theorem 1 by sketching the derivation of the following result:

Proposition 6 Let ${\phi(x) = \cos(2\pi x)}$ . If ${1/2<\gamma=1/\lambda b <1}$ and ${b\in\mathbb{N}}$ is sufficiently large, then

$\displaystyle e(1)<\gamma b$

In particular, by Corollary 3 and Tsujii’s theorem, if ${1/2<\gamma=1/\lambda b <1}$ and ${b\in\mathbb{N}}$ is sufficiently large, then Mandelbrot’s conjecture is valid, i.e.,

$\displaystyle \textrm{dim}(W_{\lambda,b}) = 2+\frac{\log\lambda}{\log b}$

Remark 11 The proof of Theorem 1 in full generality (i.e., for ${b\geq 2}$ integer and ${1/b<\lambda<1}$ ) requires the introduction of a modified version of Tsujii’s transversality condition: roughly speaking, Shen defines a function ${\sigma(q)\leq e(q)}$ (inspired from Peter-Paul inequality) and he proves

(a) a variant of Proposition 6: if ${b\geq 2}$ integer and ${1/b<\lambda<1}$ , then ${\sigma(q)<(\gamma b)^q}$ for some integer ${q}$ ;

(b) a variant of Tsujii’s theorem: if ${\sigma(q)<(\gamma b)^q}$ for some integer ${q}$ , then ${\theta\ll\textrm{Leb}_{\mathbb{R}^2}}$ .

See Sections 2, 3, 4 and 5 of Shen’s paper for more details.

We start the (sketch of) proof of Proposition 6 by recalling that the slopes of unstable manifolds are given by

$\displaystyle s(x,u):=-2\pi\sum\limits_{n=0}^{\infty} \gamma^n \sin\left(2\pi\frac{x + u_1 + u_2 b + \dots + u_n b^{n-1}}{b^n}\right)$

for ${x\in\mathbb{R}}$ , ${u\in\mathcal{A}^{\mathbb{N}}}$ , so that

$\displaystyle s'(x,u)=-4\pi^2\sum\limits_{n=0}^{\infty} \left(\frac{\gamma}{b}\right)^n \cos\left(2\pi\frac{x + u_1 + u_2 b + \dots + u_n b^{n-1}}{b^n}\right)$

Remark 12 Since ${\gamma/b < \gamma}$ , the series defining ${s'(x,u)}$ converges faster than the series defining ${s(x,u)}$ .

By studying the first term of the expansion of ${s(x,u)}$ and ${s'(x,u)}$ (while treating the remaining terms as a “small error term”), it is possible to show that if ${(k,l)\in E(1,x_0)}$ , then

$\displaystyle \left|\sin\left(2\pi\frac{x_0+k}{b}\right) - \sin\left(2\pi\frac{x_0+l}{b}\right)\right| \leq\frac{2\gamma}{1-\gamma} \ \ \ \ \ (2)$

and

$\displaystyle \left|\cos\left(2\pi\frac{x_0+k}{b}\right) - \cos\left(2\pi\frac{x_0+l}{b}\right)\right| \leq \frac{2\gamma}{b-\gamma} \ \ \ \ \ (3)$

(cf. Lemma 3.2 in Shen’s paper).

Using these estimates, we can find an upper bound for ${e(1)}$ as follows. Take ${x_0\in\mathbb{R}/\mathbb{Z}}$ with ${e(1)=e(1,x_0)}$ , and let ${k\in\mathcal{A}}$ be such that ${(k,l_1),\dots,(k,l_{e(1)})\in E(1,x_0)}$ distinct elements listed in such a way that

$\displaystyle \sin(2\pi x_i)\leq \sin(2\pi x_{i+1})$

for all ${i=1,\dots,e(1)-1}$ , where ${x_i:=(x_0+l_i)/b}$ .

From (3), we see that

$\displaystyle \left|\cos\left(2\pi x_i\right) - \cos\left(2\pi x_{i+1}\right)\right| \leq \frac{4\gamma}{b-\gamma}$

for all ${i=1,\dots,e(1)-1}$ .

Since

$\displaystyle (\cos(2\pi x_i)-\cos(2\pi x_{i+1}))^2 + (\sin(2\pi x_i)-\sin(2\pi x_{i+1}))^2 = 4\sin^2(\pi(x_i-x_{i+1}))\geq 4\sin^2(\pi/b),$

it follows that

$\displaystyle |\sin(2\pi x_i)-\sin(2\pi x_{i+1})|\geq \sqrt{4\sin^2\left(\frac{\pi}{b}\right) - \left(\frac{4\gamma}{b-\gamma}\right)^2} \ \ \ \ \ (4)$

Now, we observe that

$\displaystyle \sqrt{4\sin^2\left(\frac{\pi}{b}\right) - \left(\frac{4\gamma}{b-\gamma}\right)^2} > \frac{4}{b} \ \ \ \ \ (5)$

for ${b}$ large enough. Indeed, this happens because

${\sqrt{z^2-w^2}>2(z-w)}$ if ${z+w>4(z-w)}$ ;
${z+w>4(z-w)}$ if ${z/w:=u < 5/3}$ ;
${\frac{2\sin(\frac{\pi}{b})}{\frac{4\gamma}{b-\gamma}}\rightarrow \frac{2\pi}{4\gamma} (< \frac{5}{3})}$ as ${b\rightarrow\infty}$ , and ${2\sin(\frac{\pi}{b}) - \frac{4\gamma}{b-\gamma} \rightarrow (2\pi-4\gamma)\frac{1}{b} (>\frac{2}{b})}$ as ${b\rightarrow\infty}$ (here we used ${\gamma<1}$ ).