Hausdorff dimension of the graphs of the classical Weierstrass functions

Abstract

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

Appendix: A proof of Ledrappier’s theorem

This appendix is devoted to a proof of Ledrappier’s theorem under a further assumption that $\phi '$ is continuous (for simplicity). The proof is motivated by the original proof given in [10] and also the recent paper [9].

Let $b\ge 2$ be an integer and $\lambda \in (1/b,1)$ and let $f(x)=\sum _{n=0}^\infty \lambda ^n \phi (b^n x)$. We use z to denote a point in $\mathbb {R}^2$ and B(z, r) denote the open ball in $\mathbb {R}^2$ centered at z and of radius r. We assume that $\dim (m_x)=1$ holds for Lebesgue a.e. $x\in [0,1)$, which means for Lebesgue a.e. $x\in [0,1)$ and $\mathbb {P}$-a.e. $\mathbf u \in \mathcal {A}^{\mathbb {Z}^+}$,

$$\begin{aligned} \lim _{r\rightarrow 0} \frac{\log {\mathbb {P}}\left( \{\mathbf{v }: |S(x,\mathbf{u })-S(x,\mathbf{v })|\le r\}\right) }{\log r}=1. \end{aligned}$$

Let $\mu$ be the pushforward of the Lebesgue measure on [0, 1) under the map $x\mapsto (x, f(x))$. Let

$$\begin{aligned} \underline{d}(\mu , z)=\liminf _{r\rightarrow 0}\frac{\log \mu (B(z,r))}{\log r} \end{aligned}$$

and

$$\begin{aligned} \underline{D}=\text {essinf}\,\, \underline{d} (\mu , z). \end{aligned}$$

We shall prove that

$$\begin{aligned} \underline{D}\ge D=2+\frac{\log \lambda }{\log b}. \end{aligned}$$

This is enough to conclude that the Hausdorff dimension of the graph of f is D. Indeed, by the mass distribution principle, it implies that the Hausdorff dimension is at least D. On the other hand, it is easy to check that f is a $C^{2-D}$ function which implies that the Hausdorff dimension is at most D (see for example Theorem 8.1 of [5]).

1.1 Telescope

Define $\Phi : [0,1)\times \mathbb {R}\rightarrow [0,1)\times \mathbb {R}$ as

$$\begin{aligned} \Phi (x,y)=(bx \mod 1, (y-\phi (x))/\lambda ). \end{aligned}$$

Define $G: [0,1)\times \mathbb {R}\times \mathcal {A}^{\mathbb {Z}^+}\rightarrow [0,1)\times \mathcal {A}^{\mathbb {Z}^+}$ as

$$\begin{aligned} G(x,y, \mathbf u )=(\Phi (x,y), u_0\mathbf u ), \text { if } bx\in [u_0, u_0+1). \end{aligned}$$

The graph of f is an invariant repeller of the expanding map $\Phi$. We shall use neighborhoods bounded by unstable manifolds. For each $z_0=(x_0,y_0)\in \mathbb {R}^2$ and $\mathbf u \in \mathcal {A}^{\mathbb {Z}^+}$, let $\ell _{z_0,\mathbf u }(x)$ denote the unique solution of the initial value problem:

$$\begin{aligned} y'=-\gamma S(x,\mathbf u ), y(x_0)=y_0. \end{aligned}$$

These curves are strong unstable manifolds of $\Phi$ and they satisfy the following property: for $z=(x,y),z_0=(x_0,y_0)\in [u_0/b, (u_0+1)/b)\times \mathbb {R}$, $u_0\in \mathcal {A}$,

$$\begin{aligned} \Phi (x, \ell _{z_0, \mathbf u }(x))=(bx-u_0, \ell _{\Phi (z_0), u_0\mathbf u }(bx-u_0)). \end{aligned}$$

For $z_0=(x_0,y_0)\in [0,1)\times \mathbb {R}$, $\mathbf u \in \mathcal {A}^{\mathbb {Z}^+}$ and $\delta _1,\delta _2>0$, let

$$\begin{aligned} Q(z_0, \mathbf u ,\delta _1,\delta _2)=\{(x,y): x\in [0,1), |x-x_0|\le \delta _1, |y-\ell _{z_0,\mathbf u }(x)|\le \delta _2\}. \end{aligned}$$

The following observation was taken from [9].

Lemma 5.17

(Telescope) Let $\{(z_i,\mathbf u _i)\}_{i=0}^n$ be a G-orbit and let $x_i$ denote the first coordinate of $z_i$. For any $\delta _1,\delta _2>0$, if $\delta _1\le x_n<1-\delta _1$, then

$$\begin{aligned} \mu (Q(z_0,\mathbf u , \delta _1 b^{-n}, \delta _2 \lambda ^n))=b^{-n} \mu (Q(z_n,\mathbf u _n, \delta _1,\delta _2)). \end{aligned}$$

Proof

Let $J_i=[x_i-\delta _1 b^{i-n}, x_i+\delta _1 b^{i-n}]$, $Q_i=Q(z_i,\mathbf u _i, \delta _1 b^{i-n},\delta _2 \lambda ^{n-i})$ and let $E_i=\{x\in J_i: (x, f(x))\in Q_i\}$. Then $\mu (Q_i)=|E_i|$. Under the assumption $\delta _1\le x_n<1-\delta _1$, $Q_0$ is mapped onto $Q_n$ diffeomorphically by $\Phi ^n$. Thus $J_0$ is mapped onto $J_n$ and $E_0$ is mapped onto $E_n$ diffeomorphically by the linear map $x\mapsto b^n x$. Thus $|E_0|=b^{-n}|E_n|$.

$\square$

1.2 A version of Marstrand’s estimate

Fix a constant $t\in (1/(1+\alpha ),1)$.

Proposition 5.18

For $\mu \times \mathbb {P}$-a.e. $(z_0,\mathbf u )$,

$$\begin{aligned} \liminf _{r\rightarrow 0}\frac{\log \mu (Q(z_0,\mathbf u , r^t, r))}{\log r} \ge 1+t(\underline{D}-1). \end{aligned}$$

(6.1)

Proof

It suffices to prove that for each $\xi >0$ and $\eta >0$, there is a subset $\Sigma$ of $[0,1)\times \mathbb {R}\times \mathcal {A}^{\mathbb {Z}^+}$ with $(\mu \times \mathbb {P})(\Sigma )>1-\eta$ such that

$$\begin{aligned} \liminf _{r\rightarrow 0} \frac{\log \mu (Q(z_0,\mathbf u , r^t, r))}{\log r} \ge 1+t(\underline{D}-1)-3\xi \end{aligned}$$

(6.2)

holds for all $(z_0,\mathbf u )\in \Sigma$. By Egoroff’s theorem, we can choose $\Sigma$ with $(\mu \times \mathbb {P})(\Sigma )>1-\eta$ for which there is $r_0>0$ such that for each $(z_0,\mathbf u )\in \Sigma$,

• (S1) $\mathbb {P}\left( \left\{ \mathbf v : |S(x_0,\mathbf u )-S(x_0,\mathbf v )|\le r\right\} \right) \le r^{1-\xi }$ for each $0< r\le r_0$, where $x_0$ is the first coordinate of $z_0$;

• (S2) $\mu (B(z_0, r))\le r^{\underline{D}-\xi }$ for each $0<r\le r_0$.

In the following we shall prove that for $r>0$ small enough,

$$\begin{aligned} \int _\mathbf{u : (z_0,\mathbf u )\in \Sigma } \mu (Q(z_0, \mathbf u , r^t, r))d\mathbb {P}\le r^{1+t(\underline{D}-1)-2\xi }, \end{aligned}$$

(6.3)

holds for every $z_0\in [0,1)\times \mathbb {R}$. This is enough to conclude the proof. Indeed, let $\tau \in (0,1)$ be an arbitrary constant. Then by (6.3), there is N such that for $n>N$,

$$\begin{aligned} \mathbb {P}\left( \left\{ \mathbf u : (z_0,\mathbf u )\in \Sigma , \mu (Q(z_0,\mathbf u ,\tau ^{nt}, \tau ^n))> (\tau ^n)^{1+t(\underline{D}-1)-3\xi }\right\} \right) \le \tau ^{n\xi } \end{aligned}$$

holds for every $z_0\in [0,1)\times \mathbb {R}$. By Fubini’s theorem, this implies that

$$\begin{aligned} \mu \times \mathbb {P}\left( \left\{ (z_0,\mathbf u )\in \Sigma : \mu (Q(z_0,\mathbf u ,\tau ^{nt}, \tau ^n))> (\tau ^n)^{1+t(\underline{D}-1)-3\xi }\right\} \right) \le \tau ^{n\xi }. \end{aligned}$$

By Borel–Cantelli, it follows that for almost every $(z_0,\mathbf u ) \in \Sigma$, $\mu (Q(z_0,\mathbf u ,\tau ^{nt},\tau ^n))\le (\tau ^n)^{1+t(\underline{D}-1)-3\xi }$ holds for all n large enough. The inequality (6.2) follows.

Let us now prove (6.3). We first prove

Claim Provided that $r>0$ is small enough, for every $z_0, z\in [0,1)\times \mathbb {R}$, we have

$$\begin{aligned} \mathbb {P}(\left\{ \mathbf u : (z_0,\mathbf u )\in \Sigma , z\in Q(z_0,\mathbf u , r^t, r)\right\} )\le C_1 \left( \frac{r}{|z-z_0|}\right) ^{1-2\xi }, \end{aligned}$$

(6.4)

where $C_1>0$ is a constant.

To prove this claim, let $z=(x,y)$, $z_0=(x_0,y_0)$ and $h(x)=\ell _{z_0,\mathbf u }(x)$. Then h(x) is $C^{1+\alpha }$ with uniformly bounded norm. So

$$\begin{aligned} |y-y_0+\gamma S(x_0,\mathbf u ) (x-x_0)|\le & {} |y-h(x)|+|h(x)-h(x_0)-h'(x_0)(x-x_0)|\\\le & {} r+|\int _{x_0}^x (h'(s)-h'(x_0))ds |\le r+ C r^{t(1+\alpha )}< 2r, \end{aligned}$$

provided that r is small enough. Thus

$$\begin{aligned} \{S(x_0,\mathbf u ):(z_0,\mathbf u )\in \Sigma , z\in Q(z_0, \mathbf u , r^t, r)\} \end{aligned}$$

is contained in an interval of length $2r/(\gamma |x-x_0|)$. Since

$$\begin{aligned} |z-z_0|\le |x-x_0|+|y-y_0|\le (1+\gamma |S(x_0,\mathbf u )|)|x-x_0|+2r, \end{aligned}$$

and $|S(x,\mathbf u )|$ is uniformly bounded, the inequality (6.4) follows from the property (S1). Note that if $2r/(\gamma |x-x_0|)>r_0$, then $r/|z-z_0|$ is bounded away from zero, so (6.4) holds for sufficiently large $C_1$, since the left hand side of this inequality does not exceed one.

We continue the proof of (6.3). Note that there is a constant $C_2>0$ such that for every $r>0$ and any $z_0\in [0,1)\times \mathbb {R}$,

$$\begin{aligned} \bigcup _\mathbf{u \in \mathcal {A}^{\mathbb {Z}^+}} Q(z_0, \mathbf u , r^t, r)\subset B(z_0, C_2r^t). \end{aligned}$$

Of course we may assume there is $\mathbf u$ such that $(z_0,\mathbf u )\in \Sigma$. Thus for $R>0$ small enough, we may apply (S2) and obtain

$$\begin{aligned} \int _{B(z_0,R)} \frac{d\mu (z)}{|z-z_0|^{1-2\xi }}&=\sum _{n=0}^\infty \int _{e^{-n-1}R\le |z-z_0|< e^{-n} R} \frac{d\mu (z)}{|z-z_0|^{1-2\xi }}\\&\le \sum _{n=0}^\infty \frac{\mu (B(z_0, e^{-n}R))}{(e^{-n-1}R)^{1-2\xi }}\le \sum _{n=0}^\infty \frac{(e^{-n}R)^{\underline{D}-\xi }}{(e^{-n-1}R)^{1-2\xi }}\\&=C(\xi ) R^{\underline{D}-1+\xi }, \end{aligned}$$

where $C(\xi )$ is a constant depending on $\xi$ and $\underline{D}$. By Fubini’s theorem,

$$\begin{aligned}&\int _\mathbf{u : (z_0,\mathbf u )\in \Sigma } \mu (Q(z_0, \mathbf u , r^t, r) d\mathbb {P}(\mathbf u )\\&\quad = \int _\mathbf{u : (z_0,\mathbf u )\in \Sigma }\int _{\mathbb {R}^2} 1_{Q(z_0, \mathbf u , r^t, r)}(z) d\mu (z) d\mathbb {P}(\mathbf u )\\&\quad = \int _{B(z_0, C_2r^t)} \mathbb {P}\left( \left\{ \mathbf u : (z_0,\mathbf u )\in \Sigma , z\in Q(z_0,\mathbf u , r^t, r)\right\} \right) d\mu (z)\\&\quad \le C_1 r^{1-2\xi } \int _{B(z_0, C_2 r^t)} \frac{d\mu (z)}{|z-z_0|^{1-2\xi }} \\&\quad \le C' r^{1+t(\underline{D}-1)-(2-t)\xi }< r^{1+t(\underline{D}-1)-2\xi }, \end{aligned}$$

provided that r is small enough. $\square$

We are ready to complete the proof of Ledrappier’s theorem. For any $\xi >0$, $\eta >0$, by Proposition 5.18 and Egroff’s theorem, we can pick up a subset $\Sigma$ of $\mathbb {R}^2\times \mathcal {A}^{\mathbb {Z}^+}$ and a constant $r_*>0$ such that $(\mu \times \mathbb {P})(\Sigma )>1-3\eta$ and such that for each $(z,\mathbf u )\in \Sigma$,

$$\begin{aligned} \mu (Q(z,\mathbf u , r^t, r))\le r^{1+t(\underline{D}-1)-\xi } \text { for each } 0<r<r_*. \end{aligned}$$

We may further assume that $\Sigma \subset [\eta , 1-\eta ]\times \mathbb {R}\times \mathcal {A}^{\mathbb {Z}^+}$.

Note that $\mu \times \mathbb {P}$ is an ergodic invariant measure for the map G. By Birkhorff’s Ergodic Theorem, for almost every $(z_0,\mathbf u _0)$, there is an increasing sequence $\{n_k\}_{k=1}^\infty$ of positive integers such that $G^{n_k}(z_0,\mathbf u _0)\in \Sigma$ and

$$\begin{aligned} \liminf _{k\rightarrow \infty } n_k/n_{k+1} > 1-3\eta . \end{aligned}$$

(5.5)

For each $n=1,2,\ldots$, put $\delta _n=\gamma ^{nt/(1-t)}b^{-n}$, $r_n=\gamma ^{n/(1-t)}$, so that

$$\begin{aligned} r_n=\delta _n \lambda ^{-n},\quad \text { and }\quad r_n^t=\delta _n b^n. \end{aligned}$$

Let us prove that for k sufficiently large,

$$\begin{aligned} \frac{\log \mu (Q(z_0,\mathbf u _0, \delta _{n_k}, \delta _{n_k}))}{\log \delta _{n_k}}\ge \underline{D}+(D-\underline{D})A_1-A_2\xi , \end{aligned}$$

(5.6)

where $A_1, A_2$ are positive constants depending only on $\lambda$ and b.

Indeed, by Lemma 5.17, for k large enough,

$$\begin{aligned} \mu (Q(z_0,\mathbf u _0, \delta _{n_k}, \delta _{n_k}))=\frac{\mu (Q(G^{n_k}(z_0,\mathbf u _0),r_{n_k}^t, r_{n_k}))}{b^{n_k}} \le \frac{r_{n_k}^{1+t(\underline{D}-1)-\xi }}{b^{n_k}}. \end{aligned}$$

Using definition of $r_n$ and $\delta _n$, this gives us

$$\begin{aligned} \mu (Q(z_0,\mathbf u _0,\delta _{n_k},\delta _{n_k}))\le \delta _{n_k}^{\underline{D}}\times (b^{-n_k})^{D-\underline{D}} r_{n_k}^{-\xi }, \end{aligned}$$

Thus (5.6) holds with $A_1=\log b/(\log b+t\log \gamma ^{-1}/(1-t))$ and $A_2=\log \gamma /(t\log \gamma +(1-t)\log b^{-1})$.

By (5.5), for each n large enough, there is k such that $(1-3\eta )n_k<n_{k-1}<n\le n_k$. It follows that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\log \mu (Q(z_0,\mathbf u _0,\delta _n, \delta _n))}{\log \delta _n}\ge (1-3\eta ) (\underline{D}+(D-\underline{D})A_1-A_2\xi ). \end{aligned}$$

Since $\ell _{x_0,\mathbf u _0}$ is a smooth curve, there exists $\kappa \in (0,1)$ such that $Q(z_0,\mathbf u _0,\delta _k,\delta _k)$ contains $B(z_0, \kappa \delta _k)$ for each k. Therefore,

$$\begin{aligned} \underline{d}(\mu , z_0)=\liminf _{n\rightarrow \infty }\frac{\log \mu (Q(z_0,\mathbf u _0,\delta _n, \delta _n))}{\log \delta _n}\ge (1-3\eta ) (\underline{D}+(D-\underline{D})A_1-A_2\xi ). \end{aligned}$$

Since this estimate holds for $\mu$-a.e. $z_0$, we obtain

$$\begin{aligned} \underline{D}\ge (1-3\eta )\left( \underline{D}+ A_1(D-\underline{D})-A_2\xi \right) . \end{aligned}$$

As $\xi , \eta$ can be chosen arbitrarily small, we conclude

$$\begin{aligned} \underline{D}\ge \underline{D}+ A_1(D-\underline{D}), \end{aligned}$$

which means $\underline{D}\ge D$, as desired.